Krasnov - Aerodynamics 1

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micsN. F. Krasnov


Fundamentals of Theory. Aerodynamics of an Airfoil and a WingTranslated from the Russian by G. Leib

Mir Publishers Moscow

First published 1985 Revised from the 1980 Russian edition

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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 external 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 aerodynamics and the theory of flow over an airfoil and a wing, including an unsteady 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 discoveries 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 incompressible flow has been included here. This section presents important practical and methodological information on the conversion of the aerodynamic coefficients 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



craft whose motion is generally characterized by non-uniformity, and the noustationary 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 derivatives (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 derivatives 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 enginering. This information, 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 investigations, the application of the method of characteristics to three-dimensional gas flows, and experimental aerodynamics. I will be happy if study of this textbook 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


Preface Introduction Chapter 1Basic Information from Aerodynamics1.1. Forces Acting on a Moving Body Surface Force Property of Pressures in an Ideal Fluid Influence of Viscosity on the Flow of a Fluid 1.2. Resultant Force Action Components of Aerodynamic Forces and Moments Conversion of Aerodynamic Forces and '\[oments from One Coordinate System to Another 1.3. Determination of Aerodynamic Forces and Moments According to the Known Distribution of the Pressure and Shear Stress. Aerodynamic Coefficients Aerodynamic Forces and Moments and Their Coefficients 1.4. Static Equilibrium and Static Stability Concept of Equilibrium and Stability Static Longitudinal Stability Static Lateral Stability 1.5. Features of Gas Flow at High Speeds Compressibility of a Gas Heating of a Gas State of Air at High Temperatures

5 1325 25 25 26 28 36 36 40

41 41 52 52 53 57 58 58 59 65

Chapter 2Kinematics of a Fluid


2.1. Approaches to the Kinematic Investigation of a Fluid Lagrangian Approach Eulerian Approach Streamlines and Pathlines '.2. Analysis of Fluid Particle Motion 2.3. Vortex-Free Motion of a Fluid

71 7172





2.4. Continuity Equation General Form of the Equation Cartesian Coordinate System Curvilinear Coordinate System Continuity Equation of Gas Flow along a Curved Surface Flow Rate Equation 2.5. Stream Function 2.6. Vortex Lines 2.7. Velocity Circulation Concept Stokes Theorem Vortex-Induced Velocities~ 2.8. Complex Potential 2.9. Kinds of Fluid Flows Parallel Flow Two-Dimensional Point Source and Sink Three-Dimensional Source and Sink Doublet Circulation Flow (Vortex)

80 80 81 82 86 88 89 90 91 91 92 94 96 97 98 98 100 100 103 106 106 113 114 116 118 120 121 121 124 129 134 138 138 141 149 149 150 152 154

Chapter 3 Fundamentals of Fluid Dynamics

3.1. Equations of Motion of a Viscous Fluid Cartesian Coordinates Vector Form of the Equations of Motion Curvilinear Coordinates Cylindrical Coordinates Spherical Coordinates Equations of Two-Dimensional Flow of a Gas Near a Curved Surface 3.2. Equations of Energy and Diffusion of a Gas Diffusion Equation Energy Equation 3.3. System of Equations of Gas Dynamics. Initial and Boundary Conditions 3.4. Integrals of Motion for an Ideal Fluid 3.5. Aerodynamic Similarity Concept of Similarity Similarity Criteria Taking Account of the Viscosity and Heat Conduction 3.6. Isentropic Gas Flows Configuration of Gas let Flow Velocity Pressure, Density, and Temperature Flow of a Gas from a Reservoir

Chapter 4 Shock Wave Theory

4.1. Physical Nature of Shock Wave Formation 159 4.2. General Equations for a Shock 162 Oblique Shock 163 Normal Shock 168 4.3. Shock in the Flow of a Gas with



Constant Specific Heats System of Equations Formulas for Calculating the Parameters of a Gas Behind a Shock Oblique Shock Angle 4.4. Hodograph 4.5. A Normal Shock in the Flow of a Gas with Constant Specific Heats 4.6. A Shock at Hypersonic Velocities and Constant Specific Heats of a Gas 4.7. A Shock in a Flow of a Gas with Varying Specific Heats and with Dissociation and Ionization 4.8. Relaxation Phenomena Non-Equilibrium Flows Equilibrium Processes Relaxation Effects in Shock Waves

109 169" 170 176 179 184 186188

193 193 195 1962(10

Chapter 5Method of Characteristics

5.1. Equations for the Velocity Potential and Stream Function 5.2. The Cauchy Problem 5.3. Characteristics Com pa tibili t