mics 1 N. F. Krasnov Fundamentals of Theory. Aerodynamics of an Airfoil and a Wing Translated from the Russian by G. Leib Mir Publishers Moscow First published 1985 Revised from the 1980 Russian edition Ha anMUUCKOM Rablne © MSp;aTeJIhCTBO «BhICmaJI IDKOJIa», 1980 © English translation, Mir Publishers. 1985 ~mCs 1 H. OJ. KpaCHOB AapOAVlHOMVlKO YaCTb I OCHOBbl TeOpHH AspOAHHaMHKa npO~HnH H Kp~na j.13AaTenbCTBO «Bblcwafl WKOna» MOCKBa Preface Aerodynamics is the theoretical foundation of aeronautical, rocket, space, and artillery engineering and the cornerstone of the aerodynamic design of modern craft. The fundamentals of aerodynamics are used in studying the exter nal flow over various bodies or the motion of air (a gas) inside various objects. Engineering success in the fields of aviation, artillery, rocketry, space flight, motor vehicle transport, and so on, i.e. fields that pertain to the flow of air or a gas in some form or other, depends on a firm knowledge of aerodynamics. The present textbook, in addition to the general laws of flow of a fluid, treats the application of aerodynamics, chiefly in rocketry and modern high ,peed aviation. The book consists of two parts, each forming a separate vol ume. fhe first of them concerns the fundamental concepts and definitions of aero dynamics and the theory of flow over an airfoil and a wing, including an un steady flow (Chapters 1-9), while the second describes the aerodynamic design of craft and their individual parts (Chapters 10-15). The two parts are designed for use in a two-semester course of aerodynamics, although the first part can be used independently by those interested in individual problems of theoretical aerodynamics. A sound theoretical background is important to the study of any subject because creative solutions of practical problems, scientific research, and dis coveries are impossible without it. Students should therefore devote special attention to the first five chapters, which deal with the fundamental concepts and definitions of aerodynamics, the kinematics of a fluid, the fundamentals of fluid dynamics, the theory of shocks, and the method of characteristics used widely in investigating superO'onic flows. Chapters 6 and 7, which relate to the now over airfoils, are also important to a fundamental understanding of the subject. These chapters contain a fairly complete discussion of the general theory of now of a gas in two-dimensional space (the theory of two-dimensional flow). The information on the supersonic steady flow over a wing in Chapter 8 relates directly to these materials. The aerodynamic design of most modern craft is based on studies of such flow. One of the most topical areas of mOllern aerodynamic research is the study of optimal aerodynamic confIgurations of craft and their separate (isolated) parts (the fuselage, wing, empennage). Therefore, a small section (6.5\ that defmes a flIlite-span wing with the most advantageous planform in an incompres sible flow has been included here. This section presents important practical and methodological information on the conversion of the aerodynamic coeffic ients of a wing from one aspect ratio to another. The study of non-stationary gas flows is a rather well developed field of modern theoretical and practical aerodynamics. The results of this study are widely used in calculating the effect of aerodynamic forces and moments on 6 Preface craft whose motion is generally characterized by non-uniformity, and the nou stationary aerodynamic characteristics thus calculated are used in the dynamics of craft when studying their flight stability. Chapter 9 concerns the general relations of the aerodynamic coefficients in unsteady flow. Aerodynamic deriv atives (stability derivatives) are analysed, as is the concept of dynamic stability. Unsteady flow over a wing is also considered. The most important section of this chapter is devoted to numerical methods of calculating the stability deriv atives of a lifting surface of arbitrary planform, generally with a curved leading edge (i.e. with variable sweep along the span). Both exact and approximate methods of determining the non-stationary aerodynamic characteristics of a wing are given. A special place in the book is devoted to the most important theoretical and applied problems of high-speed aerodynamics, including the thermodynamic and kinetic parameters of dissociating gases, the equations of motion and energy, and the theory of shocks and its relation to the physicochemical properties of gases at Iitigh temperatures. Considerable attention is givl~n to shock waves (shocks), which are a manifestation of the specific properties of supersonic flows. The concept of the thickness of a shock is discussed, and the book includes graphs of the functions characterizing changes in the parameters of a gas as it passes through a shock. Naturally, a tElxtbook cannot reflect the entire diversity of problems facing the science of aerodynamics. I have tried to provide the scientific information for specialists in the field of aeronautic and rocket engine€ring. This inform ation, if mastered in its entirety, should be sufficient for young specialists to cope independently with other practical aerodynamic problems that may appear. Among these problems, not reflected in the book, are magnetogasdynamic invest igations, the application of the method of characteristics to three-dimensional gas flows, and experimental aerodynamics. I will be happy if study of this text book leads students to a more comprehensive, independent investigation ot modern aerodynamics. The book is the result of my experience teaching courses in aerodynamics at the N.E. Bauman Higher Engineering College in l\[oscow, USSR. Intended for college and junior-college students, it will also be a useful aid to specialists in research institutions, design departments. and industrial enterprises. All physical quantities are given according to the International System of Units (SI). In preparing the third Russian edition of the book, which the present English edition has been translated from, I took account of readers' remarks and of the valuable suggestions made by the reviewer, professor A.I\!. Mkhiteryan, to whom I express my profound gratitude. Nikolai F. Krasnov Contents Preface 5 Introduction 13 Chapter 1 25 Basic Information 1.1. Forces Acting on a Moving Body 25 Surface Force 25 from Aerodynamics Property of Pressures in an Ideal Fluid 26 Influence of Viscosity on the Flow of a Fluid 28 1.2. Resultant Force Action 36 Components of Aerodynamic Forces and Moments 36 Conversion of Aerodynamic Forces and '\[oments from One Coordinate System to Another 40 1.3. Determination of Aerodynamic For ces and Moments According to the Known Distribution of the Pressure and Shear Stress. Aerodynamic Coef- ficients 41 Aerodynamic Forces and Moments and Their Coefficients 41 1.4. Static Equilibrium and Static Stab- ility 52 Concept of Equilibrium and Stability 52 Static Longitudinal Stability 53 Static Lateral Stability 57 1.5. Features of Gas Flow at High Speeds 58 Compressibility of a Gas 58 Heating of a Gas 59 State of Air at High Temperatures 65 Chapter 2 2.1. Approaches to the Kinematic Invest r igation of a Fluid 71 Kinematics of a Fluid Lagrangian Approach 71 Eulerian Approach 72 Streamlines and Pathlines 73 '.2. Analysis of Fluid Particle Motion 74 2.3. Vortex-Free Motion of a Fluid 79 8 Contents 2.4. Continuity Equation 80 General Form of the Equation 80 Cartesian Coordinate System 81 Curvilinear Coordinate System 82 Continuity Equation of Gas Flow along a Curved Surface 86 Flow Rate Equation 88 2.5. Stream Function 89 2.6. Vortex Lines 90 2.7. Velocity Circulation 91 Concept 91 Stokes Theorem 92 Vortex-Induced Velocities~ 94 2.8. Complex Potential 96 2.9. Kinds of Fluid Flows 97 Parallel Flow 98 Two-Dimensional Point Source and Sink 98 Three-Dimensional Source and Sink 100 Doublet 100 Circulation Flow (Vortex) 103 Chapter 3 3.1. Equations of Motion of a Viscous Fluid 106 Fundamentals Cartesian Coordinates 106 of Fluid Dynamics Vector Form of the Equations of Motion 113 Curvilinear Coordinates 114 Cylindrical Coordinates 116 Spherical Coordinates 118 Equations of Two-Dimensional Flow of a Gas Near a Curved Surface 120 3.2. Equations of Energy and Diffusion of a Gas 121 Diffusion Equation 121 Energy Equation 124 3.3. System of Equations of Gas Dyna- mics. Initial and Boundary Con- ditions 129 3.4. Integrals of Motion for an Ideal Fluid 134 3.5. Aerodynamic Similarity 138 Concept of Similarity 138 Similarity Criteria Taking Account of the Viscosity and Heat Con- duction 141 3.6. Isentropic Gas Flows 149 Configuration of Gas let 149 Flow Velocity 150 Pressure, Density, and Temperature 152 Flow of a Gas from a Reservoir 154 Chapter 4 4.1. Physical Nature of Shock Wave For- mation 159 Shock Wave Theory 4.2. General Equations for a Shock 162 Oblique Shock 163 Normal Shock 168 4.3. Shock in the Flow of a Gas with Contents 9> Constant Specific Heats 109 System of Equations 169" Formulas for Calculating the Param- eters of a Gas Behind a Shock 170 Oblique Shock Angle 176 4.4. Hodograph 179 4.5. A Normal Shock in the Flow of a Gas with Constant Specific Heats 184 4.6. A Shock at Hypersonic Velocities and Constant Specific Heats of a Gas 186 4.7. A Shock in a Flow of a Gas with Vary- ing Specific Heats and with Dissocia- tion and Ionization 188 4.8. Relaxation Phenomena 193 Non-Equilibrium Flows 193 Equilibrium Processes 195 Relaxation Effects in Shock Waves 196 Chapter 5 5.1. Equations for the Velocity Potential and Stream Function 2(10 Method 5.2. The Cauchy Problem 205 of Characteristics 5.3. Characteristics 209 Com pa tibili ty Conui tions 209 Determination of Characteristics 209 Orthogonali ty of Characteristics 213 Transformation of the Equations for Characteristics in a Hodograph 214 Equations for Characteristics in a Hodograph for Particular Cases of Gas Flow 219 5.4. Outline of Solution of Gas-Dynamic Problems According to the !\Iethod of Characteristics 222 5.5. Application of the .'IIethod of Charac teristics to the Solution of the Prob lem on Shaping the :'\Jozzles of Super- sonic Wind Tunnels 230 Chapter 6 6.1. Thin Airfoil in all Incompressible Flow 234 Airfoil and 6.2. Transverse Flo\\' over a Thin Plate 240 Finite-Span Wing 6.3. Thin Plate at an ,\ngle of Attack 243 in an Incompressible 6.4. Finite-Span Wing in an Incompres- sible Flow 249 Flow 6.5. Wing with Optimal Planform 258 Conversion of Coefficients clJ and ex,i from One \Ving Aspect Ratio to Another 258 Chapter 7 7.1. Subsonic Flow over a Thin .. \ irioil 264 Linearization of the Equation for An Airfoil in a the Velocity Potential 264 Compressible Flow Relation Between the Parameters of a Compressible and Incompres:-ible Fluid Flow over a Thin Airfoil 266 7.2. Khristianovich .'Ilethod 269 Content of the !\Iethod 269 10 Contents Conversion of the Pressure Coefficient for an Incompressible Fluid to the Number Moo> 0 271 Conversion of the Pressure Coefficient from M001 > 0 to M002 > 0 272 Determina tion of the Cri tical Num- ber M 273 Aerodynamic Coefficients 274 7.3. Flow at Supercritical Velocity over an Airfoil (Moo> M oo,cr) 274 7.4. Supersonic Flow of a Gas with Con- stant Specific Heats over a Thin Plate 278 7.5. Parameters of a Supersonic Flow over an Airfoil with an Arbitrary Con figuration 285 Use of the Method of Characteristics 285 Hypersonic Flow over a Thin Airfoil 291 Nearly Uniform Flow over a Thin Airfoil 293 Aerodynamic Forces and Their Coefficients 293 7.6. Sideslipping Wing Airfoil 29g, Definition of a Sideslipping Wing 299 Aerodynamic Characteristics of a Sideslipping Wing Airfoil 301 Suction Force 3(5 Chapter 8 8.1. Linearized Theory of Supersonic Flow over a Finite-Span Wing 308 A Wing in Linearization of the Equation for the a Supersonic Flow Potential Function 308 Boundarv Conditions 310 Components of the Total Values of the Veloci ty Potentials and Aero- dynamic Coefficients 313 Features of Supersonic Flow over Wings 315 8.2. :Method of Sources 317 8.3. Wing wi th a Symmetric Airfoil and Triangular Planform (a = 0, c = 0) 321 Ya Flow over a Wing Panel with a Sub- sonic Leading Edge 321 Triangular Wing Symmetric about the x-Axis with Subsonic Leading Edges 326 Semi-Infinite Wing with a Supersonk Edge 328 Triangular Wing Symmetric about the x-Axis with Supersonic Leading Edges 330 8.4. Flow over a Tetragonal Symmetrie Airfoil Wing with Subsonic Edges at a Zero Angle of IAttack 331 8.5. Flow over a Tetragonal Symmetrie Airfoil Wing with Edges~of Different Kinds (Subsonic and Supersonic) 343