Theoretical and Applied Aerodynamics - Springer978-94-017-9825-9/1.pdf · Aerodynamics has a long history and it has reached a mature status during the last century. There are at

Theoretical and Applied Aerodynamics

J.J. Chattot • M.M. Hafez

Theoretical and AppliedAerodynamicsand Related Numerical Methods

123

J.J. ChattotDepartment of Mechanicaland Aerospace Engineering

University of CaliforniaDavis, CAUSA

M.M. HafezDepartment of Mechanicaland Aerospace Engineering

University of CaliforniaDavis, CAUSA

ISBN 978-94-017-9824-2 ISBN 978-94-017-9825-9 (eBook)DOI 10.1007/978-94-017-9825-9

Library of Congress Control Number: 2015932665

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Preface

The purpose of this book is to expose students to the classical theories of aero-dynamics to enable them to apply the results to a wide range of projects, fromaircraft to wind turbines and propellers. Most of the tools are analytical, butcomputer codes are also available and are used by the students to carry out seven toeight projects during the course of a quarter. These computer tools can be found athttp://mae.ucdavis.edu/chattot/EAE127/ along with the project statements.

The main focus is on aircraft and the theories and codes that help in estimatingthe forces and moments acting on profiles, wings, wing-tail and fuselage configu-rations, appropriate to the flow regime, i.e., subsonic, transonic, supersonic, viscousor inviscid, depending on the Mach number and Reynolds number.

The book culminates with a study of the longitudinal equilibrium of a glider andits static stability, a topic that is not usually found in an aerodynamics but in astability and controls book. This chapter reflects the expertise of one of the authors(JJC), who has been involved for several years in the SAE Aero Design Westcompetition, as faculty advisor for a student team, (http://students.sae.org/competitions/aerodesign/west/) and has developed the tools and capabilitiesenabling students to develop their own designs and perform well in the competition.As all airplane modelers know, placing the center of gravity in the correct locationis critical to the viability of an aircraft, and a statically stable remote controlledmodel is a requirement for human piloting.

The material is presented in a progressive way, starting with plane, two-dimensional flow past cylinders of various cross sections and then by mid-quarter,moving to three-dimensional flows past finite wings and slender bodies. In a similarfashion, inviscid incompressible flow is followed by compressible flow and tran-sonic flow, the latter requiring the numerical solution of the nonlinear transonicsmall disturbance equation (TSD). Viscous effects are discussed and also, due tononlinear governing equations, numerical simulation is emphasized.

A set of problems with solutions is placed in Part III. It corresponds to finalexaminations given over the last 10 years or so that the students have 2 hours tocomplete.

vii

Finally, the reader is assumed to have the basic knowledge in fluid mechanicsthat can be found in standard textbooks on this topic, in particular as concerns thephysical properties of fluids (density, pressure, temperature, equation of state,viscosity, etc.) and the conservation theorems using control volumes. The reader isalso assumed to master undergraduate mathematics (calculus of single variables,vector calculus, linear algebra, and differential equations). Three appendices areincluded in the book, summarizing the material relevant to the subject of interest.

Aerodynamics has a long history and it has reached a mature status during thelast century. There are at least 20 books written on aerodynamics in the last 20 years(see references). Some of these are excellent textbooks and some are outdated or outof print. All of the existing texts are based, however, on small disturbance theories.These theories are essential to gain understanding of the physical phenomenainvolved and the corresponding structure of the flow fields. They also provide goodapproximations for some simple cases. For practical problems, however, there is ademand for accurate solutions using modern computer simulation. Small distur-bance theories can still provide special solutions to test the computer codes. Moreimportant perhaps, they can provide a guideline to construct accurate and efficientalgorithms for practical flow simulations. They are also used to develop the far fieldbehavior required for the numerical solution of the boundary value problems. Ingeneral, the linearized boundary conditions and the restriction to Cartesian grids areno longer sufficient. Grid generation algorithms for complete airplanes, althoughstill a major task in a simulation, are nowadays used routinely in industry. Hence,small disturbance approximations are no longer necessary and indeed full nonlinearpotential flow codes, developed over the last two decades, are available everywhere.While it is argued that the corrections to potential flow solutions due to vorticitygenerated at the shocks can be ignored for cruising speed at design conditions, theviscous effects are definitely important to assess. Again, boundary layer approxi-mations can be useful as a guideline to construct effective viscous/inviscid inter-action procedures.

In the book we adopt this view in contrast to a complete CFD approach based onthe solution of the Navier-Stokes equations everywhere in the field for more thanone reason: it is more attractive, from an educational viewpoint, to use potentialflow model and viscous correction. It is also more practical, since Euler and henceNavier-Stokes codes are more expensive and subject to errors due to artificialviscosity as a result of the discrete approximations. A simple example is theaccurate capturing of the wake of a wing and the calculation of induced drag, still achallenge today; for the same reasons, the simulations of propellers and helicopterrotor flows are in continuing development, let alone, the problem of turbulence.

In the text, the formulation and the numerics are developed progressively toallow for both small disturbances and full nonlinear potential flows with viscous/inviscid interactions. Only a few existing books (two or three) address these issuesand we hope to cover this material in a thorough and simple manner.

viii Preface

The book contains an extensive list of references on aerodynamics includingtextbooks, advanced and specialized books, classical and old books, flightmechanics books as well as references cited in the text.

Davis, California J.J. ChattotNovember 2013 M.M. Hafez

Preface ix

Contents

Part I Fundamental Aerodynamics

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Discussion of Mathematical Models. . . . . . . . . . . . . . . . . . . 61.3 Description of the Book Content . . . . . . . . . . . . . . . . . . . . . 9References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Inviscid, Incompressible Flow Past Circular Cylindersand Joukowski Airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 152.1.4 Other Formulations . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Elementary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 Source and Sink . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.3 Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Potential Vortex. . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Superposition of Elementary Solutions . . . . . . . . . . . . . . . . . 212.3.1 Global Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Example of Superposition: Semi-infinite

Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Flow Past a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . 222.5 Flow Past Arbitrary Airfoils . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 Kutta-Joukowski Lift Theorem . . . . . . . . . . . . . . . 252.5.2 The d’Alembert Paradox . . . . . . . . . . . . . . . . . . . . 29

2.6 The Kutta-Joukowski Condition . . . . . . . . . . . . . . . . . . . . . 30

xi

2.7 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7.1 Center of Pressure—Aerodynamic Center . . . . . . . . 312.7.2 Results for the Circular Cylinder . . . . . . . . . . . . . . 31

2.8 Special Cases of Joukowski Airfoils . . . . . . . . . . . . . . . . . . 332.8.1 The Ellipse at Zero Incidence . . . . . . . . . . . . . . . . 332.8.2 The Ellipse at Incidence . . . . . . . . . . . . . . . . . . . . 352.8.3 The Flat Plate at Incidence . . . . . . . . . . . . . . . . . . 372.8.4 Circular Arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.8.5 Joukowski Airfoil at Incidence . . . . . . . . . . . . . . . 44

2.9 Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Inviscid, Incompressible Flow Past Thin Airfoils . . . . . . . . . . . . . 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Definition of a Thin Airfoil . . . . . . . . . . . . . . . . . . 513.1.2 Profile at Incidence . . . . . . . . . . . . . . . . . . . . . . . 523.1.3 Examples of Camber and Thickness

Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 Small Disturbance Linearization Method . . . . . . . . . . . . . . . 55

3.2.1 Linearization of the Tangency Condition. . . . . . . . . 563.2.2 Linearization of the Pressure Coefficient . . . . . . . . . 57

3.3 Decomposition into Symmetric and Lifting Problems . . . . . . . 583.4 The Symmetric Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Lifting Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5.1 Solution of the Fundamental Integral Equation . . . . 653.5.2 Example: Flat Plate . . . . . . . . . . . . . . . . . . . . . . . 673.5.3 Example: Parabolic Plate . . . . . . . . . . . . . . . . . . . 683.5.4 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.5 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.6 Center of Pressure . . . . . . . . . . . . . . . . . . . . . . . . 733.5.7 Aerodynamic Center. . . . . . . . . . . . . . . . . . . . . . . 743.5.8 Example of Design Problem . . . . . . . . . . . . . . . . . 75

3.6 A Family of Profiles with Minimum Pressure Gradient . . . . . 773.7 Numerical Solution of the Fundamental Integral Equation. . . . 823.8 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Inviscid, Compressible Flow Past Thin Airfoils . . . . . . . . . . . . . . 914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3 Linearized Compressible Flow Potential Equation . . . . . . . . . 934.4 Prandtl-Glauert Transformation . . . . . . . . . . . . . . . . . . . . . . 95

xii Contents

4.5 Linearized Supersonic Flow . . . . . . . . . . . . . . . . . . . . . . . . 964.5.1 Jump Conditions: Shock and Expansion Waves . . . . 1004.5.2 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.5.3 Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.5.4 Center of Pressure . . . . . . . . . . . . . . . . . . . . . . . . 1034.5.5 Aerodynamic Center. . . . . . . . . . . . . . . . . . . . . . . 104

4.6 Limit of Validity of Linearized Theories . . . . . . . . . . . . . . . 1044.7 Transonic Small Disturbance Theory . . . . . . . . . . . . . . . . . . 106

4.7.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . 1064.7.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 1104.7.3 Jump Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1104.7.4 Riemann Invariants . . . . . . . . . . . . . . . . . . . . . . . 1134.7.5 Forces and Moment . . . . . . . . . . . . . . . . . . . . . . . 1134.7.6 Murman-Cole Scheme . . . . . . . . . . . . . . . . . . . . . 1184.7.7 A Useful Nozzle Flow Solution . . . . . . . . . . . . . . . 1214.7.8 Supersonic Flow Adjacent to Uniform

Flow Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.8 Summary of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5 Inviscid, Unsteady Flows Past Airfoils . . . . . . . . . . . . . . . . . . . . . 1355.1 Unsteady Incompressible Flows. . . . . . . . . . . . . . . . . . . . . . 135

5.1.1 Unsteady Flow Past Thin Cambered Plates:Governing Equations . . . . . . . . . . . . . . . . . . . . . . 135

5.1.2 Unsteady Flow Past Thin Cambered Plates:Forces and Moment . . . . . . . . . . . . . . . . . . . . . . . 137

5.1.3 Unsteady Flow Past Thin Airfoils: Far FieldCondition for Potential . . . . . . . . . . . . . . . . . . . . . 140

5.1.4 Example: Plunging Plate . . . . . . . . . . . . . . . . . . . . 1415.1.5 Example: Pitching NACA0012 . . . . . . . . . . . . . . . 144

5.2 Unsteady Compressible Flows. . . . . . . . . . . . . . . . . . . . . . . 1455.2.1 The Full Potential Equation. . . . . . . . . . . . . . . . . . 1455.2.2 The Transonic Small Disturbance Equation . . . . . . . 148

5.3 Summary of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.4.1 Motion of a Vortex Pair Above Ground . . . . . . . . . 1515.4.2 Falling Plate over Flat Surface. . . . . . . . . . . . . . . . 152

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6 Flow Past Large and Moderate Aspect Ratio Wings . . . . . . . . . . . 1556.1 Wing Geometric Parameters . . . . . . . . . . . . . . . . . . . . . . . . 1566.2 Small Disturbance Theories . . . . . . . . . . . . . . . . . . . . . . . . 1586.3 Flow Past Thin Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Contents xiii

6.4 Fundamental Feature of the Flow Past Finite Wings:The Vortex Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.5 Two-Dimensional and Three-DimensionalVorticity Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.6 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656.6.1 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.6.2 Drag. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.7 Prandtl Lifting Line Theory (Incompressible Flow) . . . . . . . . 1726.7.1 Induced Drag for Non-ideal Wings. . . . . . . . . . . . . 1766.7.2 Wing Lift Curve and Drag Polar . . . . . . . . . . . . . . 1766.7.3 Design of an Ideal Wing. . . . . . . . . . . . . . . . . . . . 1786.7.4 Local and Global Lift Coefficients . . . . . . . . . . . . . 1816.7.5 Pitching Moment . . . . . . . . . . . . . . . . . . . . . . . . . 1836.7.6 Example of Wing Loading with Upwash. . . . . . . . . 1846.7.7 Extension of the Theory to Non-straight

Lifting Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.7.8 Numerical Solution of Prandtl

Integro-Differential Equation . . . . . . . . . . . . . . . . . 1886.8 Incompressible Flow Over Moderate Aspect Ratio Wings:

The Vortex Lattice Method. . . . . . . . . . . . . . . . . . . . . . . . . 1956.9 Compressible Flow Over Moderate Aspect Ratio Wings. . . . . 197

6.9.1 Symmetric Problem in Subsonic Flow . . . . . . . . . . 1976.9.2 Lifting Problem in Subsonic Flow . . . . . . . . . . . . . 2006.9.3 Extended Lifting Line Theory . . . . . . . . . . . . . . . . 201

6.10 Supersonic Flow Over Moderate Aspect Ratio Wings . . . . . . 2016.10.1 Symmetric Problem . . . . . . . . . . . . . . . . . . . . . . . 2016.10.2 Lifting Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2026.10.3 Rectangular Wings . . . . . . . . . . . . . . . . . . . . . . . . 203

6.11 General Wings in Subsonic and Supersonic Flows. . . . . . . . . 2056.11.1 Wings in Subsonic Flows . . . . . . . . . . . . . . . . . . . 2056.11.2 Wings in Supersonic Flows . . . . . . . . . . . . . . . . . . 2076.11.3 Wings in Transonic Flows . . . . . . . . . . . . . . . . . . 209

6.12 Transonic Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . 2106.13 Summary of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2146.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.14.1 Analysis Problem. . . . . . . . . . . . . . . . . . . . . . . . . 2156.14.2 Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 215

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

7 Axisymmetric Flows and Slender Body Theories . . . . . . . . . . . . . 2177.1 Governing Equations in Cylindrical Coordinates . . . . . . . . . . 2177.2 Small Disturbance Theory . . . . . . . . . . . . . . . . . . . . . . . . . 220

7.2.1 Calculations of Wave Drag . . . . . . . . . . . . . . . . . . 2247.2.2 Optimum Shapes . . . . . . . . . . . . . . . . . . . . . . . . . 225

xiv Contents

7.3 Lift and Induced Drag of a Body of Revolutionat Angle of Attack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7.4 Low Aspect Ratio Flat Wings . . . . . . . . . . . . . . . . . . . . . . . 2337.5 Swept and Oblique Wings . . . . . . . . . . . . . . . . . . . . . . . . . 2367.6 Wing-Body Combinations . . . . . . . . . . . . . . . . . . . . . . . . . 2367.7 Slender Bodies with General Cross Sections . . . . . . . . . . . . . 2387.8 Supersonic Area Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2437.9 Conical Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.9.1 Method of Taylor and Maccoll . . . . . . . . . . . . . . . 2467.9.2 Small Disturbance Approximations

for Flows Over a Cone . . . . . . . . . . . . . . . . . . . . . 2487.9.3 Conical Flow Past a Delta Wing . . . . . . . . . . . . . . 2507.9.4 Rectangular Wings at Angle of Attack . . . . . . . . . . 2547.9.5 Numerical Solution of Conical Euler Equations . . . . 254

7.10 Summary of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2567.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

8 Viscous Fluid Flow and Laminar Boundary Layers . . . . . . . . . . . 2618.1 Incompressible 2-D Flows . . . . . . . . . . . . . . . . . . . . . . . . . 261

8.1.1 Vorticity Versus Strain Rate in 2-D . . . . . . . . . . . . 2628.1.2 Viscous Stresses in 2-D (Cartesian Coordinates) . . . 2648.1.3 Constitutive Relations. . . . . . . . . . . . . . . . . . . . . . 2668.1.4 Navier-Stokes Equations for 2-D

Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . 2668.1.5 Laminar Boundary Layer Theory (Prandtl 1904) . . . 2678.1.6 Boundary Layer over a Flat Plate. . . . . . . . . . . . . . 2698.1.7 Numerical Method for the Solution of Boundary

Layer Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 2708.1.8 Boundary Layer Thicknesses . . . . . . . . . . . . . . . . . 271

8.2 Compressible Viscous Fluid Flow . . . . . . . . . . . . . . . . . . . . 2728.2.1 Viscous Stresses and Constitutive Relations . . . . . . 2728.2.2 Navier-Stokes Equations for 2-D

Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . 2728.2.3 Energy Equation for Compressible

Viscous Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . 2738.2.4 Boundary Layer Equations for Compressible

Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2748.2.5 Further Simplifications for the Case Pr ¼ 1. . . . . . . 2758.2.6 Determination of Drag . . . . . . . . . . . . . . . . . . . . . 2758.2.7 Unsteady Boundary Layer. . . . . . . . . . . . . . . . . . . 276

8.3 Historical and Classical Works . . . . . . . . . . . . . . . . . . . . . . 2778.3.1 Blasius Solution for the Flat Plate

(No Pressure Gradient) . . . . . . . . . . . . . . . . . . . . . 277

Contents xv

8.3.2 Flow Past a Wedge (Falkner/Skan). . . . . . . . . . . . . 2788.3.3 von Karman Integral Momentum Equation . . . . . . . 2798.3.4 Derivation of von Karman Integral Equation

for Compressible Flows with Pressure Gradient . . . . 2818.3.5 Transformations of Boundary Layer Equations . . . . 2838.3.6 Flow Separation. . . . . . . . . . . . . . . . . . . . . . . . . . 2868.3.7 Flow at the Trailing Edge . . . . . . . . . . . . . . . . . . . 2888.3.8 Three Dimensional Boundary Layers . . . . . . . . . . . 289

8.4 Summary of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2908.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

9 Viscous/Inviscid Interaction Procedures . . . . . . . . . . . . . . . . . . . . 2939.1 Viscous/Inviscid Interaction Procedures Based

on Displacement Thickness Concept . . . . . . . . . . . . . . . . . . 2939.1.1 Coupling Integral Formulations of both Inviscid

and Viscous Flow Regions . . . . . . . . . . . . . . . . . . 2979.1.2 Coupling the Numerical Solution of the Partial

Differential Equations of the Inviscid Flowwith Integral Equations of Boundary Layers . . . . . . 303

9.1.3 Coupling the Numerical Solution of the PartialDifferential Equations of the Viscous Flowwith the Integral Equations of the OuterInviscid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

9.1.4 Coupling Numerical Solutions of PartialDifferential Equations of both the Inviscidand Viscous Flow Regions . . . . . . . . . . . . . . . . . . 315

9.2 Viscous/Inviscid Interaction Procedures Basedon Domain Decomposition Techniques . . . . . . . . . . . . . . . . 315

9.3 Summary of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

Part II Special Topics

10 Wind Turbine and Propeller Aerodynamics—Analysisand Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32710.1 Introduction—the Different Types of Wind Turbines . . . . . . . 327

10.1.1 Aerodynamics Forces—Lift and Drag. . . . . . . . . . . 32810.1.2 Savonius Rotor . . . . . . . . . . . . . . . . . . . . . . . . . . 33010.1.3 Darrieus Rotor. . . . . . . . . . . . . . . . . . . . . . . . . . . 33010.1.4 Horizontal Axis Wind Turbine. . . . . . . . . . . . . . . . 333

10.2 General 1-D Conservation Theorems—ActuatorDisk Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

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10.3 Vortex Model and Strip Theory—The Goldstein Model . . . . . 33610.3.1 General Comments. . . . . . . . . . . . . . . . . . . . . . . . 33610.3.2 Adimensionalization—Discretization

of the Vortex Sheets. . . . . . . . . . . . . . . . . . . . . . . 33810.3.3 Biot-Savart Law—Induced Velocities . . . . . . . . . . . 34110.3.4 Forces and Moment . . . . . . . . . . . . . . . . . . . . . . . 34210.3.5 Wake Equilibrium Condition . . . . . . . . . . . . . . . . . 344

10.4 Aerodynamic Design of a Rotor Blade—BetzMinimum Energy Condition . . . . . . . . . . . . . . . . . . . . . . . . 34510.4.1 Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34510.4.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 34810.4.3 Viscous Correction. . . . . . . . . . . . . . . . . . . . . . . . 351

10.5 Analysis of the Flow Past a Given Rotor . . . . . . . . . . . . . . . 35310.5.1 Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35310.5.2 Algorithm for High Incidences—Regularization

of the Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 35410.5.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

10.6 Unsteady Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 35710.6.1 Effect of Yaw . . . . . . . . . . . . . . . . . . . . . . . . . . . 35910.6.2 Tower Interference Model . . . . . . . . . . . . . . . . . . . 36010.6.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

10.7 Hybrid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36310.8 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36610.9 Propellers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36710.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

11 Glider and Airplane Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37311.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37311.2 Model of a Classical Wing/Tail Configuration. . . . . . . . . . . . 374

11.2.1 Linear Model for the Main Wing . . . . . . . . . . . . . . 37411.2.2 Linear Model for the Tail . . . . . . . . . . . . . . . . . . . 37511.2.3 Aerodynamic Coefficients for the Fuselage . . . . . . . 37711.2.4 Global Aerodynamic Coefficients for the Glider. . . . 378

11.3 Rapid Prototyping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37911.4 Acceleration Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38211.5 Longitudinal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 38311.6 Static Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38611.7 Winglet Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39011.8 Trimming the Glider for Maximum Distance. . . . . . . . . . . . . 39211.9 Trimming the Glider for Maximum Duration . . . . . . . . . . . . 39311.10 Classical Versus Canard Configurations . . . . . . . . . . . . . . . . 393

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11.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39511.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

12 Introduction to Hypersonic Flows . . . . . . . . . . . . . . . . . . . . . . . . 39912.1 Regimes of Compressible Flow . . . . . . . . . . . . . . . . . . . . . . 39912.2 Inviscid Theories of Hypersonic Flows. . . . . . . . . . . . . . . . . 401

12.2.1 Hypersonic Potential Flow . . . . . . . . . . . . . . . . . . 40112.2.2 Prandtl/Meyer Expansion for Hypersonic Flows. . . . 40312.2.3 Hypersonic Flow over a Flat Plate

at Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . 40412.2.4 Power Series Expansion of Pressure Coefficients . . . 40412.2.5 Bodies of Revolution . . . . . . . . . . . . . . . . . . . . . . 40512.2.6 Hypersonic Small Disturbance Theory . . . . . . . . . . 40612.2.7 Unsteady Analogy . . . . . . . . . . . . . . . . . . . . . . . . 40912.2.8 Unsteady Hypersonic Flows . . . . . . . . . . . . . . . . . 41012.2.9 Unified Supersonic-Hypersonic Small

Disturbance Theory . . . . . . . . . . . . . . . . . . . . . . . 41112.2.10 Formulation of the Full and Reduced Problems

in Terms of Stream Functions . . . . . . . . . . . . . . . . 41112.2.11 Similarity Solutions for Power Law Bodies . . . . . . . 41612.2.12 Newtonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . 41912.2.13 Cole’s Slender Body Theory of Newtonian Flow . . . 42212.2.14 Slender Body at Angle of Attack, Bow Shocks

Around Blunt Bodies and Numerical Simulationsof Hypersonic Inviscid Flows . . . . . . . . . . . . . . . . 426

12.3 Viscous Hypersonic Flows . . . . . . . . . . . . . . . . . . . . . . . . . 43212.3.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . 43312.3.2 Laminar Boundary Layer in Weak

Interaction Regime . . . . . . . . . . . . . . . . . . . . . . . . 43512.3.3 Solutions of Laminar Boundary Layer

Equations at Hypersonic Speeds . . . . . . . . . . . . . . 43812.3.4 Weak Viscous/Inviscid Interactions . . . . . . . . . . . . 44012.3.5 Strong Viscous/Inviscid Interaction . . . . . . . . . . . . 44112.3.6 Theoretical Developments . . . . . . . . . . . . . . . . . . . 443

12.4 Hypersonic Area Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44412.5 Hypersonic Similitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44612.6 Hypersonic Vehicle Design. . . . . . . . . . . . . . . . . . . . . . . . . 447References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

13 Flow Analogies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45713.1 Hele-Shaw Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45713.2 Hydraulic Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

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13.3 Electric Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46813.3.1 Analog Representation of Circulation

Around Lifting Airfoils. . . . . . . . . . . . . . . . . . . . . 46913.3.2 Analog Study of Flows Around Bodies

of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 47013.3.3 Hodograph Tank . . . . . . . . . . . . . . . . . . . . . . . . . 47013.3.4 Analog Study of Supersonic Conical Flows. . . . . . . 471

13.4 Analog Study of Three-Dimensional Flows. . . . . . . . . . . . . . 47213.5 Sobieczky’s Rheograph-Transformations. . . . . . . . . . . . . . . . 47413.6 Electronic Analog Computers: Networks Versus Tanks . . . . . 47513.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

Part III Problems and Solutions

14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48114.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

14.1.1 Thin Airfoil Theory (2-D Inviscid Flow). . . . . . . . . 48114.1.2 Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . 48214.1.3 Equilibrium of the Glider

(3-D Incompressible Flow) . . . . . . . . . . . . . . . . . . 48314.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

14.2.1 Thin Airfoil Theory (2-D Inviscid Flow). . . . . . . . . 48414.2.2 Lifting Line Theory (3-D Inviscid Flow). . . . . . . . . 48414.2.3 Airplane Longitudinal Equilibrium . . . . . . . . . . . . . 485

14.3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48614.3.1 Thin Airfoil Theory (2-D Inviscid Flow). . . . . . . . . 48614.3.2 Lifting Line Theory (3-D Inviscid Flow). . . . . . . . . 48714.3.3 Glider Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 487

14.4 Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48814.4.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 48814.4.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 49014.4.3 Equilibrium of the Aggie Micro Flyer . . . . . . . . . . 490

14.5 Problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49214.5.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 49214.5.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 49414.5.3 Equilibrium of the Aggie Micro Flyer . . . . . . . . . . 494

14.6 Problem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49614.6.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 49614.6.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 49714.6.3 Equilibrium of the Aggie Micro Flyer . . . . . . . . . . 498

14.7 Problem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49914.7.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 499

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14.7.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 50014.7.3 Equilibrium of the AMAT09 . . . . . . . . . . . . . . . . . 501

14.8 Problem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50214.8.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 50214.8.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 50414.8.3 Equilibrium of the AMAT10 . . . . . . . . . . . . . . . . . 505

14.9 Problem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50614.9.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 50614.9.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 50814.9.3 Equilibrium of the AMAT11 . . . . . . . . . . . . . . . . . 509

14.10 Problem 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51014.10.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 51014.10.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 51114.10.3 Equilibrium of the Glider . . . . . . . . . . . . . . . . . . . 513

15 Solutions to Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51515.1 Solution to Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

15.1.1 Thin Airfoil Theory (2-D Inviscid Flow). . . . . . . . . 51515.1.2 Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . 51715.1.3 Equilibrium of the Glider

(3-D Incompressible Flow) . . . . . . . . . . . . . . . . . . 51815.2 Solution to Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

15.2.1 Thin Airfoil Theory (2-D Inviscid Flow). . . . . . . . . 52015.2.2 Lifting Line Theory (3-D Inviscid Flow). . . . . . . . . 52115.2.3 Airplane Longitudinal Equilibrium . . . . . . . . . . . . . 523

15.3 Solution to Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52415.3.1 Thin Airfoil Theory (2-D Inviscid Flow). . . . . . . . . 52415.3.2 Lifting Line Theory (3-D Inviscid Flow). . . . . . . . . 52615.3.3 Glider Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 528

15.4 Solution to Problem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52915.4.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 52915.4.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 53115.4.3 Equilibrium of the Aggie Micro Flyer . . . . . . . . . . 533

15.5 Solution to Problem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53415.5.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 53415.5.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 53815.5.3 Equilibrium of the Aggie Micro Flyer (AMF III) . . . 540

15.6 Solution to Problem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54215.6.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 54215.6.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 54515.6.3 Equilibrium of the Aggie Micro Flyer . . . . . . . . . . 547

xx Contents

15.7 Solution to Problem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54815.7.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 54815.7.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 55215.7.3 Equilibrium of the AMAT09 . . . . . . . . . . . . . . . . . 553

15.8 Solution to Problem 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55515.8.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 55515.8.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 55915.8.3 Equilibrium of the AMAT2010 . . . . . . . . . . . . . . . 561

15.9 Solution to Problem 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56215.9.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 56215.9.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 56715.9.3 Equilibrium of the AMAT11 . . . . . . . . . . . . . . . . . 568

15.10 Solution to Problem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57015.10.1 2-D Inviscid, Linearized, Thin Airfoil Theories . . . . 57015.10.2 Prandtl Lifting Line Theory. . . . . . . . . . . . . . . . . . 57315.10.3 Equilibrium of the Glider . . . . . . . . . . . . . . . . . . . 575

Appendix A: Special Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

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