Pilot's Operating Handbook

3.1 Airspeed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2 Equations of Motion for Straight, Level, and Unaccelerated Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 Thrust and Power Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4 Takeoff and Landing Performance . . . . . . . . . . . . . . . . . . . . 91 3.5 Gliding Fight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.6 Climbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.7 Endurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.8 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.9 Turn Performance and V-n Diagrams. . . . . . . . . . . . . . . . . . . 127 3.10 Historical Snapshot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 ix Chapter 4 Aircraft Equations of Motion . . . . . . . . . . . . . . . . . . . 145 4.1 Aircraft Axis Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.3 Aircraft Force Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.4 Moment Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.5 Longitudinal and Lateral-Directional Equations of Motion . . . . . 164 4.6 Kinematic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.7 Historical Snapshot—Genesis 2000 Flight Simulator. . . . . . . . . 169 Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Chapter 5 Aircraft Static Stability . . . . . . . . . . . . . . . . . . . . . . . . 173 5.1 Static Stability Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.2 Stability, Control Power, and Cross-Control Derivatives, and Control Deflection Sign Convention . . . . . . . . . . . . . . . . . 174 5.3 Longitudinal Applied Forces and Moments . . . . . . . . . . . . . . . 177 5.4 Longitudinal Static Stability . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.5 Lateral-Directional Applied Forces and Moments . . . . . . . . . . . 202 5.6 Lateral-Directional Static Stability . . . . . . . . . . . . . . . . . . . . . 215 5.7 Summary of Steady-State Force and Moment Derivatives . . . . . 229 5.8 Historical Snapshot—The X-38 Mid-Rudder Investigation . . . . . 230 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Chapter 6 Linearizing the Equations of Motion . . . . . . . . . . . . . . 239 6.1 Small Perturbation Approach . . . . . . . . . . . . . . . . . . . . . . . . 239 6.2 Developing the Linearized Aircraft Equations of Motion . . . . . . 241 6.3 First-Order Approximation of Applied Aero Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.4 First-Order Approximation of Perturbed Thrust Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 6.5 Recasting the Equations of Motion in Acceleration Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 6.6 Historical Snapshot—The X-38 Parafoil Cavity Investigation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Chapter 7 Aircraft Dynamic Stability . . . . . . . . . . . . . . . . . . . . . 303 7.1 Mass-Spring-Damper System and Classical Solutions of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 303 7.2 Root Representation Using the Complex Plane . . . . . . . . . . . . 316 7.3 Transforming the Linearized EOM to the Laplace Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 x TABLE OF CONTENTS 7.4 Dyanmic Stability Guidelines . . . . . . . . . . . . . . . . . . . . . . . . 356 7.5 Cooper–Harper Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 7.6 Experimental Determination of Second-Order Parameters . . . . . 372 7.7 Historical Snapshot—The A-10A Prototype Flight Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 Chapter 8 Classical Feedback Control . . . . . . . . . . . . . . . . . . . . . 389 8.1 Open-Loop Systems, Transfer Functions, and Block Diagrams . . 389 8.2 Closed-Loop Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 8.3 Closed-Loop Analysis of a Second-Order System. . . . . . . . . . . 394 8.4 Closed-Loop Transfer Functions . . . . . . . . . . . . . . . . . . . . . . 399 8.5 Time Response Characteristics . . . . . . . . . . . . . . . . . . . . . . . 403 8.6 Root Locus Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 8.7 Historical Snapshot—The C-1 Autopilot . . . . . . . . . . . . . . . . . 427 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 Chapter 9 Aircraft Stability and Control Augmentation. . . . . . . . . 433 9.1 Inner-Loop Stability and Control . . . . . . . . . . . . . . . . . . . . . . 433 9.2 Outer-Loop Autopilot=Navigation Control . . . . . . . . . . . . . . . . 444 9.3 Compensation Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 9.4 Combined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 9.5 Historical Snapshot—The A-7D DIGITAC Digital Multimode Flight Control System Program . . . . . . . . . . . . . . . 463 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 Chapter 10 Special Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 10.1 System Type and Steady-State Error . . . . . . . . . . . . . . . . . . . 473 10.2 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 10.3 Digital Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 10.4 Advanced Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . 549 10.5 Reversible and Irreversible Flight Control Systems . . . . . . . . . 551 10.6 Spins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 10.7 Historical Snapshot—The F-16 Fly-by-Wire System . . . . . . . . 558 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 Appendix A Conversions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 Appendix B Properties of the U.S. Standard Atmosphere. . . . . . . . 573 Appendix C Airfoil Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 Appendix D T-38 Performance Data . . . . . . . . . . . . . . . . . . . . . . 589 TABLE OF CONTENTS xi Appendix E Selected Laplace Transforms. . . . . . . . . . . . . . . . . . . 603 Appendix F Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Appendix G Development of Longitudinal and Lateral-Directional Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . 609 Appendix H Stability Characteristics of Selected Aircraft . . . . . . . 613 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Series listing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631 xii TABLE OF CONTENTS Preface This textbook was created as a resource for teaching aircraft performance, static stability, dynamic stability, and classical feedback control as part of an undergraduate aeronautical engineering curriculum. Chapters 1 through 5 are intended for a one-semester course in performance and static stability, while Chapters 6 through 9 are intended for a sequential one-semester course in dynamic stability and feedback control. The text is intended to provide an understandable first exposure to these topics as well as a logical progression of subject matter. These courses are normally taken during the junior year follow- ing a fundamental course in aeronautics. This text in draft form was used as the course text for the first two courses in aircraft flight mechanics at the U.S. Air Force Academy during a four-year period preceding publication. The experience and student feedback obtained was used to improve and expand the text. The text was also used at the Air Force Academy for an undergraduate aeronautical engineering elective course in aircraft feedback control systems, normally taken after completion of the aircraft dynamic stability and feedback control course. Chapters 6 through 9 were covered at a fairly rapid pace and Chapter 10 provided new material and additional depth. This text may also serve as a reference for the practicing engineer. Thomas R. Yechout January 2003 xiii This page intentionally left blank Acknowledgments Many individuals contributed to the development of this textbook. First, I would like to thank Col. Michael L. Smith and Col. D. Neal Barlow, the past and present heads of the Department of Aeronautics at the U. S. Air Force Academy, for their encouragement and support throughout the five years required to bring this effort from concept to reality. Special thanks are due to Lt. Col. (Ret.) Steven L. Morris for his contributions to Chapters 2 and 4 and for his many years of contributing to the development of flight mechanics courses at the Air Force Academy. Thanks also to Col. (Ret.) Wayne F. Hallg- ren for his contributions to Chapters 1 and 3 and to Lt. Col. David E. Bossert for his contributions to Chapter 10. Excellent contributions and review were provided by Bill Blake and Dave Leggett of the Air Force Research Labora- tory, Flight Vehicles Directorate, Dr. Jeff Ashworth of Embry Riddle Aeronau- tical University, Dr. Dennis Bernstein of the University of Michigan, and Meredith Cawley of the AIAA staff. Thanks are also due to Lt. Col. Dave Bossert, Lt. Col. Steve Pluntze, Col. (Ret.) Gene Rose, Capt. Alex Sansone, Lt. Col. Scott Wells, and Dr. Tom Cunningham, all of the Air Force Academy Department of Aeronautics, for the time devoted to providing detailed review comments. In addition, AIAA reviewers and numerous Air Force Academy cadets majoring in aeronautical engineering provided review comments. Finally, I would like to thank my wife, Kathy, for her continued support throughout the development of this textbook and the career that I love. Thomas R. Yechout xv 1 A Review of Basic Aerodynamics Lift, drag, thrust, and weight are the four primary forces acting on an aircraft in flight (refer to Fig. 1.1). Lift and drag are ‘‘aerodynamic forces’’ arising because of the relative motion between the aircraft and the surrounding air. Thrust is provided by the propulsive system, and the force due to gravity is called ‘‘weight.’’ Ultimately, we want to adequately predict an aircraft’s motion. An under- standing of lift, drag, and thrust is essential to this end. This chapter provides the basics of lift and drag, while Chapter 2 introduces propulsion. These chap- ters are not designed to replace an aerodynamics or propulsion course, but do provide a baseline we can build on. 1.1 Fundamental Concepts and Relationships We will begin our discussion with a review of fundamental aerodynamic concepts and relationships. A sound understanding of these concepts is neces- sary to establish a solid foundation for the study of aircraft flight mechanics. 1.1.1 Properties of a Flowfield and a Discussion of Units The study of aerodynamics deals with the flow of air. As a body moves through air, or any fluid (liquid or gas) for that matter, the surrounding air is disturbed. The term ‘‘flowfield’’ is common in the language of aerodynamics and is used to refer to the air in the vicinity of the body. Pressure, density, temperature, and velocity are the key physical properties of aerodynamics. A goal of the aeronautical engineer is to quantify these prop- erties at every point in the flowfield. We will begin by defining each of these properties: 1) Pressure ( p) ‘‘is the normal force per unit area exerted on a surface due to the time rate of change of momentum of the gas molecules impacting the surface’’ (Ref. 1). At sea level, atmospheric pressure is approximately 2116 psf. Pressure distributions on an aircraft, caused by the same physical mechanism (namely an exchange of momentum between air molecules and a body) will be discussed elsewhere in this book. 2) The density ( r) of air is its mass (weight=acceleration due to gravity) per unit volume. A high-density flow implies closely compacted air molecules. 3) Temperature (T ) is a measure of the average kinetic energy of the air molecules. A high temperature indicates that the air molecules are moving randomly at relatively high speeds. 1 4) Velocity (V ) is a vector quantity; it has both magnitude and direction. The velocity at any point in the flowfield is the velocity of an infinitesimally small fluid element (differential ‘‘chunk’’ of air) as it sweeps through that point. The English Engineering System is used in this text. Based on a consistent set of units (from Newton’s 2nd law), this system is typically chosen in the study of flight mechanics. Assuming constant mass, Newton’s 2nd law is: F ¼ ma A pound force (lb) is defined as the force necessary to accelerate one slug (our unit of mass) one foot, per second squared. Table 1.1 displays the dimensions and units used for our fundamental properties. Consider a flowfield as shown in Fig. 1.2. Our four properties are called ‘‘point properties.’’ In general, they vary from point to point within the flow- field. Additionally, these properties can be a function of time; this is called ‘‘unsteady’’ flow. Pressure can be a function of not only location, but also of time, for example p ¼ pðx; y; z; tÞ. A ‘‘steady flow’’ assumption removes the time dependency; therefore, p ¼ pðx; y; zÞ. Obviously, this makes our analysis more simple. For the case of Table 1.1 Dimensions and units used in this book PROPERTY DIMENSIONS UNITS Pressure ( p) force=area lb=ft2 (psf) Density (r) mass=volume slug=ft3 Temperature (T ) n=a deg Rankine (R) Velocity (V ) length=time ft=s (fps) Fig. 1.1 Simplified illustration of the four forces acting on an aircraft. 2 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS straight, level, and unaccelerated flight, steady flow is a reasonable assumption. It would be unreasonable to assume steady flow for a rapid pitch-up maneuver. Another important concept, which relates to velocity, is the definition of a ‘‘streamline.’’ A streamline is a curve that is tangent to the velocity vectors in a flow. For example, refer to Fig. 1.3. Consider two arbitrary streamlines in the flow, as shown in Fig. 1.4. A consequence of the definition of a streamline in steady flow is that the mass flow (slug=s) passing through cross section 1 must be the same as that passing through 2. By definition, there is no mechanism for mass to cross a stream- line—the mass flow rate must be conserved between the two streamlines. Shortly, the significance of this will become more clear. For flow over an airfoil, the distance between streamlines is decreased as they pass over and above the airfoil. As we will see, this indicates an increase in velocity. Fig. 1.2 Point in a flowfield. Fig. 1.4 Two streamlines in a flowfield. Fig. 1.3 Streamline in a flowfield. A REVIEW OF BASIC AERODYNAMICS 3 1.1.2 Equation of State for a Perfect Gas A perfect gas assumes that intermolecular forces are negligible. For the pressures and densities characteristic of flight mechanics applications, this assumption is extremely reasonable. The equation governing a perfect gas is: p ¼ rRT where R is the specific gas constant, a function of the gas considered. For example, its value for air is different than for argon. For normal air (not, for example, chemically reacting air) the value of R is: R ¼ 1716 ft-lb ðslugÞðRÞ ½English UnitsŠ ¼ 287 J ðkgÞðKÞ ½Metric UnitsŠ Looking at the units of R, temperature must be in degrees Rankine [English Units] or Kelvin [Metric Units] to properly use the equation of state. Example 1.1 An aircraft is flying at an air pressure of 10 psi and a temperature of 20F. What is the air density for these conditions? Using the equation of state and solving for density, we have: r ¼ p RT We must next convert to consistent units. p ¼ 10 psi ¼ ð10 psiÞð144 psf =psiÞ ¼ 1440 psf T ¼ 20F ¼ 460 þ ð20 Þ ¼ 440R Finally, r ¼ 1440 ð1716Þð440Þ ¼ 0:00191slug=ft 3 1.1.3 Hydrostatic Equation Consider a differential fluid element of air shown in Fig. 1.5. Its mass is dm, and it has dimensions as shown below. In the vertical direction there are two forces—weight and the forces due to pressure acting on the top and bottom surface areas (dA). Consider a force balance in the vertical, or z, direction, SFz ¼ pdA  ð p þ dpÞdA  ðrdAdhÞg 4 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS or dp ¼ rgdh Called the hydrostatic equation because it was originally derived for water, this differential equation relates a change in h, or altitude, with a change in pres- sure. Note, a positive increase in altitude corresponds to a negative change in pressure. As altitude increases, pressure decreases. Integrating between heights 1 and 2 in the vertical direction yields ð2 1 dp ¼  ð2 1 rgdh Assuming constant r and g, p2  p1 ¼ rgðh2  h1Þ ð1:1Þ This relationship is known as the manometry equation and is valid for a fluid (typically a liquid) of constant density in a uniform gravitational field. Example 1.2 A lake is 50 ft deep. What is the difference in pressure between the bottom of the lake and the surface given that the density of water is 1.94 slug=ft 3 ? Using the manometry equation [Eq. (1.1)], we have p2  p1 ¼ rgðh2  h1Þ We will designate position 1 as the surface of the water (h1 ¼ 0) and position 2 as the bottom of the lake (h2 ¼ 50 ft). We then have p2  p1 ¼ ð1:94Þð32:2Þð50  0Þ ¼ 3123:4 psf Fig. 1.5 Differential fluid element of air. A REVIEW OF BASIC AERODYNAMICS 5 Thus, the pressure at the bottom of the lake is 3123.4 psf higher than at the surface. 1.1.4 Continuity Equation The laws of aerodynamics are governed by physical principles. When these principles are applied to an appropriate model, useful equations can be derived. The continuity equation is based on the physical law that mass is conserved. Consider the ‘‘stream tube’’ in Fig. 1.6, which can be thought of as a bundle of streamlines. Let us also recall that mass cannot cross a streamline. If we assume a steady flow, such that the properties everywhere in the flow- field are time independent, then the mass flow rate across 1 and 2 must be the same. Now, we will define a one-dimensional flow, which is a flow in which the properties are assumed constant at each cross section (perpendicular to the flow’s velocity) of the flow. To help your understanding, consider Fig. 1.7 in which 1 and 2 are arbitrary points on a cross section of the flow. By assuming one-dimensional flow, we neglect any variation in the velocity across a specific cross section. The amount of incremental mass, dm, that enters the stream tube during the incremental time, dt, can be defined as dm ¼ rAdx ¼ rAV dt where A is the cross-sectional area of the stream tube. The following expres- sion then follows for the mass flow rate through the stream tube: dm dt ¼ _m m ¼ rAV Fig. 1.6 Three-dimensional stream tube. Fig. 1.7 Illustration of one-dimensional flow. 6 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS Because mass is conserved, the mass flow rate is the same at any cross section in the stream tube and the continuity equation reduces to the following simple result: r1A1V1 ¼ r2A2V2 ð1:2Þ or, rAV ¼ constant. The dimensions are mass per time. It should make sense that the mass flow rate is a function of density, velocity, and cross-sectional area. Example 1.3 Consider the following nozzle. Find the velocity V2 at the nozzle exit given that V1 ¼ 35 ft=s r1 ¼ 0:002 slug=ft3 A1 ¼ 0:5 ft 2 r2 ¼ 0:0015 slug=ft3 A2 ¼ 0:05 ft 2 Using the continuity equation [Eq. (1.2)] and solving for V2 , V2 ¼ r1A1V1 r2A2 ¼ ð0:002Þð0:5Þð35Þ ð0:0015Þð0:05Þ ¼ 466:7 ft=s 1.1.5 Incompressible and Compressible Flow Under certain conditions, it is reasonable to assume the flowfield is essen- tially incompressible, or constant density flow. This assumption is typically made for low-speed flowfields, where velocities everywhere (all x; y; z loca- tions) are less than 330 ft=s. Later we will define Mach number (M ) and note that this threshold corresponds to a Mach number of 0.3 at sea-level, standard- day conditions. Note that the continuity equation reduces to the following for the case of a one-dimensional, steady, and incompressible flow. The dimensions have, of course, changed—they are now ft 3=s, or volumetric flow rate. AV ¼ Constant A REVIEW OF BASIC AERODYNAMICS 7 As an aside, water (another fluid) is virtually incompressible. For this reason, flowfield density variations are typically ignored for water and other liquids. Example 1.4 For the nozzle of Example 1.3, assume the fluid is incompressible water and V1 remains at 35 ft=s. Find V2 and the volumetric flow rate. Using the incompressible form of the continuity equation and solving for V2 , we have V2 ¼ V1A1 A2 ¼ ð35Þð0:5Þ 0:05 ¼ 350 ft=s The volumetric flow rate would be, volumetric flow rate ¼ V1A1 ¼ V2A2 ¼ ð35Þð0:5Þ ¼ 17:5 ft 3=s 1.1.6 Bernoulli’s Equation Newton’s 2nd law is used again (refer to the hydrostatic equation) to derive another extremely useful equation. Consider a differential fluid element moving along a streamline, as shown in Fig. 1.8. In general, the forces acting on the fluid element are (refer to Fig. 1.9) 1) Weight, or force due to gravity 2) Normal forces (pressure times surface area) acting on all six sides 3) Tangential forces due to the friction between adjacent fluid elements For the purpose of this discussion, only the forces in the streamline direc- tion are shown. In fact, forces due to pressure and friction act on all six surfaces. Assume a steady flow and neglect the weight of the fluid element—in essence, we are assuming the fluid element’s weight is small (for air) in comparison to the pressure forces ‘‘pushing’’ the element along the streamline. Furthermore, neglect the effects of friction. By applying Newton’s 2nd law, Fig. 1.8 Differential fluid element moving along a streamline. 8 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS F ¼ ma reduces to the following differential equation (for details, refer to Ref. 2), called Euler’s equation: dp ¼ rV dV Euler’s equation is not convenient for easily solving problems. If we make one more assumption, incompressible flow, density is assumed constant, and the equation is easily integrated between two points along a streamline. ð2 1 dp ¼  ð2 1 rV dV p2  p1 ¼ r V 2 2 2  V 2 1 2   p1 þ 1 2 rV 2 1 ¼ p2 þ 1 2 rV 2 2 ð1:3Þ or p þ 1 2rV 2 ¼ constant ðalong a streamlineÞ static pressure dynamic pressure ðqqÞ As long as the assumptions are valid, the summation of static pressure ( p) and dynamic pressure (qq ¼ 1 2 rV 2 ) remains constant along the streamline. Frequently, ‘‘total pressure’’ ( p0 ) is used to identify the constant, or p0 ¼ p þ qq ¼ total pressure Equation (1.3) is called Bernoulli’s equation or the momentum equation and is one of the classics of aerodynamics. The equation is algebraic. Remember, this only applies for incompressible flow. Additionally, the four assumptions behind Euler’s equation are still buried in the result—the equation is applied along a streamline, steady flow is assumed, and forces due to weight and friction are neglected. Fig. 1.9 Forces acting on a fluid element.   A REVIEW OF BASIC AERODYNAMICS 9 Let’s pause for a moment. Note that both the continuity and the momentum equations relate properties (pressure, density, and velocity) between points in a flow, say A and B. On the other hand, the equation of state can only be applied at a single point; it says nothing about how the properties at point B relate to the properties at point A. Example 1.5 An F-15 on approach to Tyndall Air Force Base is flying at 120 kn. The atmospheric pressure and density are 2116 psf and 0.00238 slug=ft 3 , respec- tively. At a point on the upper surface of the wing, the pressure is measured as 2060 psf. Find the velocity of the flow at this point on the wing and the total pressure acting on the aircraft. We can use Bernoulli’s equation [(Eq. (1.3)] since the flow is incompres- sible: p1 þ 1 2 rV 2 1 ¼ p2 þ 1 2 rV 2 2 We will designate a point out in front of the aircraft as position 1 and the point on the wing as position 2. First find the total pressure based on position 1 conditions. We will convert the airspeed to consistent units. V1 ¼ 120 kn ¼ ð120 knÞ 1:69 ft=s kn   ¼ 202:8 ft=s and then find the total pressure. p0 ¼ p1 þ 1 2 rV 2 1 ¼ 2116 þ 1 2 ð0:00238Þð202:8Þ2 ¼ 2165 psf Because total pressure is constant, p0 ¼ p2 þ 1 2 rV 2 2 we can solve for V2 : V2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp0  p2Þ r s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2165  2060Þ 0:00238 r ¼ 297 ft=s 10 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 1.1.7 The Speed of Sound For a perfect gas, the speed of sound (a) is calculated from the following equation (refer to Ref. 3 for a detailed derivation): a ¼ ffiffiffiffiffiffiffiffiffi gRT p ð1:4Þ In this equation g is the ratio of specific heats. For most aerodynamic applica- tions, it is assumed to be a constant equal to 1.4 for air. Note that the speed of sound for a perfect gas is only a function of temperature. The propagation of a sound wave takes place through molecular collisions. If the air molecules are moving faster, because they are excited by high temperatures, then the speed of the sound wave is faster. Temperature must be in R to obtain a speed of sound in ft=s. Example 1.6 What is the speed of sound if the air temperature is 70F? First, we convert to absolute temperature. T ¼ 70F ¼ 70 þ 460 ¼ 530R Using Eq. (1.4), a ¼ ffiffiffiffiffiffiffiffiffi gRT p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1:4Þð1716Þð530Þ p ¼ 1128 ft=s 1.1.8 Mach Number and Aerodynamic Flight Regimes Figure 1.10 presents a representative streamline around an aircraft. At any point in the flowfield, the local Mach number (M ) can be defined as the ratio of the local flow velocity to the local speed of sound, M ¼ V a ð1:5Þ Fig. 1.10 Local Mach numbers for a streamline around an aircraft. A REVIEW OF BASIC AERODYNAMICS 11 The Mach number at point 2 is probably larger than the Mach number at point 1 because of the acceleration of the flow velocity around the contours of the aircraft. Point 1 is sufficiently far ahead of the aircraft such that the properties at that point have not been disturbed by the presence of the aircraft. With the aircraft as the reference frame, V1 becomes the aircraft’s true airspeed. The conditions at point 1 are called freestream conditions and denoted by a subscript 1. If the local speed of sound is the same as the local velocity, the Mach number is 1.0 (or ‘‘sonic’’) at that point in the flow. Figure 1.11 defines aerodynamic flight regimes based on freestream Mach number. Four regimes are defined: 1) When the local Mach number is less than 1.0 everywhere in the flowfield, the flow is ‘‘subsonic.’’ 2) When the local Mach number is greater than 1.0 everywhere in the flowfield, the flow is ‘‘supersonic.’’ 3) When the flowfield has regions of both subsonic and supersonic flow, the flowfield is ‘‘transonic.’’ Depending on airspeed and geometry, transonic flow typically occurs at ‘‘freestream’’ Mach numbers between approxi- mately 0.8 and 1.2. 4) A flow is called ‘‘hypersonic’’ when certain physical phenomena become important that were not important at lower speeds. These include, for example, high temperature effects and relatively thin shock layers. Typi- cally, Mach 5 is used as the hypersonic threshold, but this value is greatly dependent on the shape of the body of interest. Refer to Ref. 4 for more detail. The dynamic pressure, qq, may easily be defined in terms of Mach number using Eqs. (1.4), (1.5), and the equation of state for a perfect gas. qq ¼ 1 2 rV 2 ¼ 1 2 rðMaÞ2 ¼ 1 2 rM 2gRT From the equation of state, r ¼ p RT We thus have an alternate form for qq: qq ¼ 1 2 gpM 2 ð1:6Þ Fig. 1.11 Aerodynamic flight regimes. 12 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 1.2 The Standard Atmosphere A discussion of aerodynamics would not be complete without introducing the concept of a standard atmosphere. Air pressure, temperature, and density (also viscosity) are functions of altitude. The standard atmosphere, typically presented in a tabular form, assigns values to these properties as they change with altitude. It provides a common reference for Department of Defense (DoD), academia, and industry. For example, suppose the Air Force wants to purchase an interceptor. To compare climb performance (critical to an interceptor mission) between competing aircraft, manufacturers present data based on their aircraft operating on a standard day, which is an imaginary day when the pressure, temperature, and density behave exactly as defined in the standard atmosphere. Otherwise, it would be nearly impossible to accurately assess how one aircraft performs against another. The standard atmosphere was generated by starting from an assumed (easiest property to measure) temperature distribution. Figure 1.12 shows an ideal variation of temperature with altitude based on many experimental samplings. The temperature is assumed to remain constant between approxi- mately 36,000 and 82,000 ft; this is called the isothermal region. With a known temperature profile, two laws of physics (hydrostatic equation and the equation of state) were used to mathematically ‘‘build’’ the standard atmosphere. A current version of the standard atmosphere is presented in Appendix B, and a more detailed discussion of the standard atmosphere devel- opment is presented in Ref. 5. The standard atmosphere properties of tempera- ture, density, and pressure may be presented in the form of ratios, as defined in Eq. (1.7). Note: y, s, and d all have the value of 1.0 at sea level conditions: y ¼ T TSL s ¼ r rSL d ¼ p pSL ð1:7Þ Empirical equations have been developed to predict the temperature and pressure ratios as a function of altitude. These predictions are aligned with the Fig. 1.12 Standard atmosphere temperature variation. A REVIEW OF BASIC AERODYNAMICS 13 1962 U.S. Standard Atmosphere. They are divided into the altitude regions below and above approximately 36,000 ft (the troposphere region where temperature decreases at a linear rate, and the isothermal stratosphere region). For altitudes (h) less than or equal to 36,089 ft, we have y ¼ 1  6:875  106 h d ¼ ð1  6:875  106 hÞ5:2561 s ¼ d y 9 > > > = > > > ; h  36;089 ft ð1:8Þ For altitudes from 36,000 ft to approximately 65,600 ft, we have y ¼ 0:75189 d ¼ 0:2234eð4:806105 ð36;089hÞÞ s ¼ d y 9 > > > = > > > ; 36;089 ft < h < 65;600 ft ð1:9Þ The altitude (h) must be input in feet in the previous equations. The relation- ship shown for density ratio can be derived using the equation of state for a perfect gas. Frequently, in the language of flight and aeronautical engineering, the terms pressure, temperature, and density altitudes are used. Consider an aircraft flying at 10,000 ft above sea level, as shown in Fig. 1.13. For the ambient pressure and temperatures shown, we use the standard atmosphere table to find these values. The standard atmosphere altitude corre- sponding to a pressure of 1484 psf is 9500 ft, and the aircraft is said to be flying at a pressure altitude (hp) of 9500 ft. Pressure altitude says nothing about how high the aircraft is above the ground. Rather, the aircraft is ‘‘seeing’’ an air pressure as though it were flying at 9500 ft on a standard day. Similarly, a temperature altitude (h T ) can be defined. For example, with an ambient temperature of 479:5R, the aircraft is said to be at an 11,000-ft temperature altitude because 479:5R is the standard atmosphere temperature for 11,000 ft. Density altitude (hr) follows the same approach. Pressure, temperature, and density altitude relate pressure, temperature, and density, respectively, to the standard atmosphere model. Simply stated, density altitude is the standard Fig. 1.13 Aircraft at specified flight conditions. 14 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS atmosphere altitude that has the same value of density as the conditions under consideration. 1.3 Airfoil Fundamentals Figure 1.14 presents a sketch of an airfoil, or the cross section of a wing. By definition, the flow over the airfoil is assumed to vary only in the x and z (z is perpendicular to the surface) direction. The span (in the y-direction) is assumed to approach infinity. Frequently, airfoils are also called two-dimen- sional wings or infinite wings because wing-tip effects (refer to Sec. 1.4) are ignored. 1.3.1 Source of Aerodynamic Forces and Relative Motion In Fig. 1.15, an airfoil is shown in a flowfield. The only way nature can transmit an aerodynamic force is through pressure and shear stress distributions acting on the airfoil surface. Pressure and shear stress act at every point on the body, for example, points 1 and 2 in Fig. 1.15. Pressure always acts perpendi- cular to the surface. Shear stress (tw) acts tangentially to the surface (or ‘‘wall’’)—like pressure, it has the dimensions of force per unit area (refer to Sec. 1.3.3.1). The net effect is an aerodynamic force (Faero ). Later, we will break Faero into lift and drag components and consider the moment created by the pressure and shear stress distributions. Fig. 1.14 Airfoil section. Fig. 1.15 Pressure and shear stress vectors on an airfoil. A REVIEW OF BASIC AERODYNAMICS 15 Incidentally, the same is true of any body in a flowfield, be it an automobile, a ski jumper, or a cyclist—pressure and shear stress are the only physical mechanisms that generate an aerodynamic force. As you might expect, aerodynamic forces depend on the relative velocity between the body and the air. Consider two cases, as shown in Fig. 1.16. Shown is an airfoil on a test stand with air blowing over it at 300 ft=s and an identical airfoil flying at 300 ft=s through still air. The two airfoils have the same aerodynamic force, which is why wind tunnels work. 1.3.2 Lift As shown in Fig. 1.17, lift (L) and drag (D) are the components of the aero- dynamic force perpendicular and parallel, respectively, to the relative wind (V1). In this section, we will focus on lift. Recall that nature transmits an aerodynamic force through pressure and shear stress distributions. The pressure distribution over an airfoil, or wing for that matter, is primarily responsible for lift. Consider a pressure distribution as shown in Fig. 1.18. The longer arrows denote pressures higher than freestream pressure; shorter arrows are lower pressures. Simply stated, lift is generated by creating a net pressure difference between the upper and lower surfaces. As we will see later in this chapter, an airfoil’s geometry is one of the keys to efficiently generating lift. By referring to Fig. 1.19 and looking at continuity and Bernoulli’s equation, we can gain some insight into how lift is generated. Although these two equa- tions have several assumptions buried in them, they very nicely capture the basic physics to explain lift. Air must speed up to get over the curved upper surface of an airfoil. This can be viewed as an area constriction or nozzle effect, with the continuity equation predicting an increase in velocity. As the Fig. 1.16 Airfoil on a test stand and an airfoil in flight. Fig. 1.17 Lift and drag components of Faero . 16 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS air speeds up, the pressure goes down, as predicted by Bernoulli’s equation. The reduced pressure on the upper surface, relative to the higher pressure on the lower surface, creates a lift force in the upward direction. Note that the streamlines get closer (cross-sectional area goes down) as they pass over the upper surface. From continuity, we know velocity must increase to pass the mass between the streamlines; this is no different than putting your thumb over the end of a garden hose to speed up the water. From Bernoulli’s equation, we know that if velocity increases, pressure decreases. Therefore, a high velocity implies low pressure, low velocity implies high pressure, and a pressure differential between the upper and lower surfaces leads to lift. As you might expect, the amount of lift depends on a number of parameters, for exam- ple flow velocity and airfoil shape. This is discussed in subsequent sections. 1.3.3 Drag Refer again to Fig. 1.17 in which drag is the component of the aerodynamic force parallel to V1. To gain an appreciation for drag, where it comes from and its consequences, we need some more tools. We will start with the concept of a viscous flow. 1.3.3.1 Introduction to viscous flow. A viscous flow is one in which the effects of viscosity, thermal conduction, and=or mass diffusion are important. As a particle moves about in space, it carries with it its momentum, energy, and mass (its identity). Viscosity is due to the transport of momentum—it is important if a flowfield has large velocity gradients. Thermal conduction results from the Fig. 1.18 Pressure distribution over an airfoil. Fig. 1.19 Flow streamlines over an airfoil. A REVIEW OF BASIC AERODYNAMICS 17 transport of energy and similarly is significant in regions of strong temperature gradients. Mass diffusion is due to the transport of mass—it is important in regions of strong concentration gradients, for example, in a chemically reacting flowfield. For the purpose of this book, we will ignore the effects of thermal conduc- tion and mass diffusion. The airspeeds we are concerned with do not yield flowfields where these effects are important. Therefore, in this book, a viscous flow implies regions in which there are strong velocity gradients. Velocity gradients cause shear stress. Remember from Sec. 1.3.1 that pressure and shear stress distributions generate aerodynamic forces. To understand why shear stresses exist, consider a shear flow as sketched in Fig. 1.20. The streamlines are horizontal; however, velocity varies in the y direction. Therefore, a velocity gradient, in the y-direction, exists. Because there is a velocity gradient, a fluid element above the plane a–b is moving faster than a fluid element below the plane. There is a rubbing action, or friction, between the fluid elements because of the different velocities. Because of an exchange of momentum (mass times velocity) between the fluid elements, a force is exerted on the plane a–b. We give it a special name, shear stress (ta–b), where the subscript denotes the stress is acting on the plane a–b. As you might expect, shear stress (t) is proportional to the strength of the velocity gradient. The constant of proportionality is called the coefficient of viscosity and is given the symbol m. t / dV dy  ab t ¼ m dV dy  ab ð1:10Þ Physically, viscosity is a measure of a fluid’s resistance to shear and has the dimensions of mass=(length  time). The standard day sea level value for viscos- ity is m ¼ 3:7373  107 slug=ðft  sÞ Fig. 1.20 Illustration of a shear flow. 18 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS The coefficient of viscosity is a function of temperature. Qualitatively, its value changes as shown in Fig. 1.21. From the figure, it is apparent that the viscosity of air increases as tempera- ture increases. Recall, viscosity is due to the transport of momentum. Air molecules have more velocity and, hence, more momentum in high-temperature flowfields. 1.3.3.2 The concept of a boundary layer. Consider the flow over an airfoil, as shown in Fig. 1.22. There is a relatively small region adjacent to the airfoil where the effects of viscosity (or friction) must be taken into account. Called a boundary layer, the concept was introduced by Ludwig Prandtl in 1904. Outside this region, the flowfield is assumed to be inviscid or frictionless. Within a boundary layer, velocity gradients are severe—the flow is retarded because of the presence of the body. From our previous discussion, this implies that viscous effects are important. As presented in Fig. 1.23, a boundary-layer profile describes how the flow velocity changes in a direction normal (in the y direction as shown) to the surface of a wing, fuselage, or any solid surface exposed to an air stream. Note that the velocity is zero at the surface, which is the so-called ‘‘no slip boundary condition.’’ Frequently, the subscript w (for ‘‘wall’’) is used to denote the surface boundary conditions. The effect of friction, between the air and the body, diminishes in the y direction. Hence, the velocity increases through the boundary layer (in the y direction) until eventually the presence of friction is Fig. 1.21 Coefficient of viscosity as a function of temperature. Fig. 1.22 Flow over an airfoil with boundary layer region. A REVIEW OF BASIC AERODYNAMICS 19 no longer felt, and the velocity gradient approaches zero. The boundary layer thickness is denoted by d. Two types of boundary layers exist: laminar and turbulent. A laminar boundary layer is characterized by smooth and regular streamlines, or smoothly layered flow. In contrast, a turbulent boundary layer is ‘‘random, irregular, and tortuous.’’7 Laminar and turbulent velocity profiles are different. To gain some insight into the differences, consider the flow over a flat plate. At some stream- wise distance, two representative velocity profiles are shown in Fig. 1.24: The turbulent profile does not imply a nice, orderly boundary layer. Rather, what is really shown is how the average velocity changes in the y direction. From the figure, note the following 1) A turbulent boundary layer is thicker than a laminar boundary layer (dturb > dlam ). For a flat plate, the freestream velocity (V1) is 99.9% recovered at the edge of the boundary layer. 2) The velocity gradient at the wall is greater for a turbulent boundary layer. For a given y distance, the turbulent velocity is greater than the laminar velocity. Fig. 1.23 Boundary-layer profile. Fig. 1.24 Laminar and turbulent boundary layer profiles. 20 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 3) The shear stress at the wall is higher for a turbulent boundary layer [twÞturb > twÞlam ]. This implies a higher skin friction drag for a turbulent boundary layer. 1.3.3.3 Reynolds number and transition. Consider the flow over a sharp flat plate as shown in Fig. 1.25; the distance ‘‘x’’ is measured from the leading edge. The Reynolds number, based on a characteristic length (in this case x), is defined as Re x ¼ r1V1x m1 ð1:11Þ Physically, the Reynolds number is a ratio of inertia forces to viscous forces— it is nondimensional. For nearly all our applications, the Reynolds number is relatively high (on the order of 10 4 10 8 ) and inertia forces dominate. The Reynolds number is useful in predicting if a boundary layer is laminar or turbulent. Transition is defined as the point (in reality a relatively small region) where the boundary layer changes from laminar to turbulent. Once again, consider the flow over a sharp flat plate as shown in Fig. 1.26. Typically, a boundary layer starts as laminar. Eventually, for a variety of reasons, it will transition to being turbulent. The distance xcrit locates the transi- tion point. In reality, there will be a transition region, where the boundary layer has pockets of both laminar and turbulent flow. A Reynolds number, based on xcrit , is defined as Re xcrit ¼ r1V1xcrit m1 ð1:12Þ or xcrit ¼ Re xcrit m1 r1V1 ð1:13Þ For the case of flow over a smooth flat plate, the critical Reynolds number is approximately 500,000. Therefore, given the freestream conditions, the critical Fig. 1.25 Flow over a flat plate. A REVIEW OF BASIC AERODYNAMICS 21 Reynolds number provides a means to predict where transition will occur, which is very handy, but very arbitrary. Various factors influence where transition takes place. For example, surface roughness will trip a boundary layer and cause it to go turbulent earlier than expected. Additionally, surface temperature, Mach number, and pressure gradi- ents all affect transition. Because of its significant ramifications, researchers will continue working to find better ways to predict transition. Example 1.7 A flat plate with a 1-ft length in a wind tunnel test section is being tested at 150 and 300 ft=s at standard sea-level conditions. If the critical Reynolds number is 500,000, find the location where the flow transitions from laminar to turbulent for each velocity. Using Eq. (1.13), we have at 150 ft=s: xcrit ¼ Re xcrit m1 r1V1 ¼ ð500;000Þð3:737  107Þ ð0:00238Þð150Þ ¼ 0:523 ft At 300 ft=s we have: xcrit ¼ Re xcrit m1 r1V1 ¼ ð500;000Þð3:737  107Þ ð0:00238Þð300Þ ¼ 0:262 ft Thus, we can see that the transition point moves forward as the velocity is increased. The 300 ft=s case is illustrated next. Fig. 1.26 Typical boundary layer transition from laminar to turbulent flow. 22 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 1.3.3.4 Skin friction drag. Because of the presence of friction between an aerodynamic body and the flowfield, two forms of drag are created: skin friction drag and pressure drag. We will discuss skin friction drag first. Consider the flowfield in Fig. 1.27. Within the boundary layer, the velocity increases from zero at the surface. This velocity gradient causes a shear stress (tw) at the surface, or wall—recall Eq. (1.10) for shear stress, which has the dimensions of force per unit area. When integrated over the entire surface area, the result is called skin friction drag and given the notation D f . The skin friction coefficient, Cf , is defined as Cf ¼ Df qq1Swet ð1:14Þ Swet is the so-called ‘‘wetted area,’’ which is a surface area that would get wet if the aerodynamic body were in water. C f is obviously going to depend on freestream conditions and boundary layer type. For the simple case of flow over a flat plate, experimental=theoretical values of skin friction coefficient are presented in the following equations: Cf ¼ 1:328 ffiffiffiffiffiffiffiffi ReL p ðlaminarÞ C f ¼ 0:074 ðReLÞ1=5 ðturbulentÞ The equations assume a fully laminar or turbulent boundary layer from the leading edge (that is, with no transition). Although simple relationships, the results give some valuable physical insight. Note the Reynolds number dependence, where L is the characteristic length (the total running length of the flat plate in this case). For a given Reynolds number, C f is greater for the turbulent case. Because there is such a close relationship between Reynolds number and boundary layer type (hence shear stress), this result is not surprising. Regardless of the aerodynamic shape, a laminar boundary layer will cause less skin friction drag than a turbulent boundary layer. D f Þturbulent > Df Þlaminar Fig. 1.27 Boundary layer velocity profile. A REVIEW OF BASIC AERODYNAMICS 23 Example 1.8 A rectangular wing has a 5-ft chord and a 40-ft span. Estimate the skin fric- tion drag acting on the wing at a velocity of 100 ft=s and sea-level conditions assuming the critical Reynolds number is 600,000. We will first calculate the transition location using Eq. (1.13). xcrit ¼ Re xcrit m1 r1V1 ¼ ð600;000Þð3:737  107Þ ð0:00238Þð100Þ ¼ 0:942 ft We will next assume turbulent flow for the entire wing and compute Df using the second equation from Table 1.2. Df ¼ C f 1 2   rV 2S Cf ¼ 0:074 Re0:2 L For the entire wing, L ¼ 5 ft, so that Re L ¼ rVL m ¼ ð0:00238Þð100Þð5Þ 3:737  107 ¼ 3:18  10 6 and S  ð5 ftÞð40 ftÞ ¼ 200 ft 2 for the upper surface of the wing Cf ¼ 0:074 ð3:18  10 6Þ0:2 ¼ 0:0037 D f turbulent wing ¼ ð0:0037Þ 1 2   ð0:00238Þð100Þ2ð200Þ ¼ 8:81 lb ðon one wing surfaceÞ However, the flow is not turbulent over the entire wing; therefore, we next calculate the turbulent skin friction drag associated with the laminar region so it can be subtracted out from the previous result. For the laminar region Re L ¼ rVL m ¼ ð0:00238Þð100Þð0:942Þ 3:737  107 ¼ 6:0  10 5 Cf ¼ 0:074 ð6  10 5Þ0:2 ¼ 0:00517 Dfturbulent forward wing ¼ ð0:00517Þ 1 2   ð0:00238Þð100Þ2ð0:942  40Þ ¼ 2:32 lb ðone surfaceÞ 24 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS The turbulent skin friction drag on the portion of the wing aft of xcrit is thus: D fturbulent aft wing ¼ Dfturbulent wing  D fturbulent forward wing ¼ 8:81  2:32 ¼ 6:49 lb ðone surfaceÞ We must next calculate the skin friction drag on the forward portion of the wing which is in laminar flow. Using the first equation from Table 1.2, C f ¼ 1:328 ffiffiffiffiffiffiffiffi ReL p ¼ 1:328 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6:0  10 5 p ¼ 0:00171 where Re L ¼ rVL m ¼ ð0:00238Þð100Þð0:942Þ 3:737  107 ¼ 6:0  10 5 The laminar skin friction drag on the upper wing surface is then, Dflaminar forward wing ¼ Cf 1 2   rV 2S ¼ 0:00171 1 2   ð0:00238Þð100Þ2ð0:94  40Þ ¼ 0:765 lb and the total skin friction drag on the upper surface is Df ¼ D flaminar forward wing þ Dfturbulent aft wing ¼ 0:765 þ 6:49 ¼ 7:255 lb For the upper and lower surface of the wing, we simply multiply by two. D fwing ¼ 2ð7:255Þ ¼ 14:53 lb A sketch (not to scale) of the wing is shown: A REVIEW OF BASIC AERODYNAMICS 25 1.3.3.5 Pressure drag. As discussed, the presence of friction leads to skin friction drag. Additionally, friction also causes another form of drag; this is called drag due to the pressure field or simply pressure drag. To understand the concept of pressure drag, consider the viscous flow over a cylinder, as sketched in Fig. 1.28—two representative streamlines are shown. Because the velocity is zero at point 1, it is called a stagnation point. Pres- sure is a maximum here—recall the inverse relationship between velocity and static pressure. Between 1 and 2, the flow accelerates to its maximum velocity and achieves a minimum pressure. Aft of point 2, the pressure begins to increase. In the meantime, friction has sufficiently reduced the flow’s energy such that it cannot overcome the increasing pressure aft of point 2. The flow separates and a wake is formed on the aft side of the cylinder. Pressure drag is created. Qualitatively, the pressure ( p) on the cylinder’s surface is shown in Fig. 1.29—j (see Fig. 1.28) is 0 deg at the stagnation point, 90 deg at Point 2. Note the difference between the high pressure on the front (pushing the cylin- der to the right) and relatively lower pressure on the back (pushing to the left). This leads to pressure drag. Physically, the same thing happens for an airfoil. Consider Fig. 1.30. Once again, the pressure is highest at the stagnation point. Over the upper surface, the velocity increases to a maximum—a minimum pressure is reached. Aft of this point, the pressure increases. When pressure increases in the streamwise direction, the pressure gradient is called adverse. In contrast, if the pressure gradient decreases in the streamwise direction, the gradient is favorable. When the flowfield has insufficient momentum (or energy) to overcome the adverse Fig. 1.28 Flow over a cylinder. Fig. 1.29 Pressure distribution on a cylinder’s surface. 26 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS pressure gradient, it separates from the upper surface and the airfoil stalls. Drag goes up; lift goes down. Because the physics of flow separation is so important, we will look at it again but from the perspective of the velocity profiles inside the boundary layer. Consider the boundary layer on the upper surface of an airfoil as shown in Fig. 1.31—x is a running length along the airfoil’s surface. We will assume the boundary layer transitions to turbulent at xcrit . Friction takes its toll, and eventually the fluid particles near the surface have insufficient energy to overcome the adverse pressure. The fluid particles slow down, and then stop—this is the separation point. The velocity gradient at the wall is zero. Downstream of this point, the fluid particles may actually backup (called flow reversal) because of an adverse pressure gradient. How can separation be delayed? We can energize the boundary layer by increasing the momentum, or kinetic energy, of the fluid elements within it. This is most easily accomplished by tripping the boundary layer to make it turbulent. Recall, a turbulent boundary layer has a larger (than laminar) velo- city gradient near the wall. Therefore, a turbulent boundary layer has more momentum and thus is able to withstand an adverse pressure gradient longer before separating. Vortex generators, and various other boundary layer control (BLC) devices, are all designed to delay separation and consequently reduce the penalties associated with pressure drag. Obviously, the stronger the adverse pressure gradient, the more susceptible an airfoil will be to flow separation and a stalled condition. Therefore, airfoil Fig. 1.30 Pressure regions on an airfoil. Fig. 1.31 Boundary layer velocity profiles. A REVIEW OF BASIC AERODYNAMICS 27 design and orientation to the freestream velocity are critical to an efficient lift- ing surface. Another type of pressure drag is referred to as base drag. Base drag is typi- cally associated with long, slender shapes, such as missiles and fuselages, which have relatively blunt aft ends. For example, a cylindrical missile of constant diameter may simply end at the rear of the missile without tapering to a point. A separated flow region will exist at the base or aft area of the missile because the flow will not be able to stay attached around the sharp corner of this base region. Base drag can account for up to 50% of the total drag on a missile or projectile. Fig. 1.32 presents a variation of base drag with Mach number for a missile with a length to diameter ratio of 7.2. 1.3.3.6 Profile drag. The combined drag because of skin friction and flowfield separation (pressure drag) is called profile drag. D ¼ D f þ D p We have arrived at one of the great compromises of aerodynamics. A laminar boundary layer decreases skin friction drag but very likely will increase pres- sure drag. A turbulent boundary layer will typically reduce pressure drag, but will increase skin friction drag. Ultimately, the shape=orientation of the body will dictate which type of drag is dominant. As you might expect, skin friction is dominant for slender bodies, while pressure drag dominates blunt or bluff bodies. Consider the flow over a sphere. Qualitatively, a graph of profile drag vs Reynolds number is shown in Fig. 1.33. Incidentally, the easiest way to change Reynolds number is to change velocity. Note the dramatic decrease in drag when the Reynolds number is sufficiently large enough to cause the laminar boundary layer to transition to turbulent. Although skin friction drag increases Fig. 1.32 Variation of base drag coefficient with Mach number for a missile shape with 7.2 fineness ratio (see Ref. 6). 28 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS with a turbulent boundary layer, pressure drag is clearly dominant for a sphere (bluff body). A turbulent boundary layer, with its higher energy flow, can better overcome the strong adverse pressure gradient, thereby reducing the overall profile drag. Example 1.9 An illustration of two spheres in flow at the same velocity is shown below. There’s only one difference between the two cases—the one on the left has a smooth surface and the one on the right has a dimpled surface to trip a turbu- lent boundary layer. Which ball has lower separation drag? Separation is significantly delayed on the ball on the right. As a result, the overall profile drag is reduced because separation drag is significantly larger than skin friction drag. For this same reason, golf balls have dimples so that the ball will travel farther in the air. In Sec. 1.3.5, we’ll examine airfoil data and reinforce the impact a boundary layer has on profile drag. 1.3.4 Airfoil Terminology Airfoils are the fundamental building block for aircraft wing and tail surface design. In this section we will discuss the definition of airfoil configurations and aerodynamic characteristics. Fig. 1.33 Drag variation with Reynolds number. A REVIEW OF BASIC AERODYNAMICS 29 1.3.4.1 Geometry and nomenclature. The concept of an airfoil (wing cross section) was introduced at the beginning of Sec. 1.3. Defining the geometry of an airfoil can get complicated (refer to Ref. 7). In this book, we will use a few common definitions to grasp the basics of airfoil geometry. Consider Fig. 1.34. Leading edge and trailing edge are self-explanatory. Other definitions follow: 1) A straight line, passing through the leading and trailing edge, is called the chord line. The straight-line distance between the leading and trailing edge is the chord. 2) The mean camber line is the locus of points halfway between the upper and lower surfaces, as measured perpendicular to the mean camber line itself— positive camber is shown (typical for wing sections). 3) The max camber (sometimes called simply camber) is the maximum distance between the mean camber line and the chord line, as measured perpendicular to the chord line. 4) The thickness is the distance between the upper surface and lower surface, as measured perpendicular to the mean camber line. 5) The angle of attack (a) is the angle between the chord line and the freestream velocity (V1), or relative wind. 6) The airfoil in Fig. 1.34 is cambered. For the case of an uncambered, or symmetric, airfoil the top and bottom surface are identical—the mean camber line is the same as the chord line. For convenience, government and industry have devised numerous ways to geometrically describe, with numbers and letters, various airfoil shapes. One extremely common airfoil designation is the National Advisory Committee for Aeronautics (NACA), a precursor to NASA, four-digit series. Four digits define an airfoil shape: the first is the amount of maximum camber in percent of chord, the second is the location of the maximum camber in tenths of chord, the last two digits are the maximum thickness of the airfoil in percent of chord. For example, a NACA 2412 airfoil would have a two percent (0.02chord) maximum camber, the maximum camber would occur at the 40% chord location (x ¼ 0.4chord), and the maximum thickness would be 12% of the chord length (0.12chord). There are several other NACA designa- Fig. 1.34 Airfoil geometry. 30 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS tions to describe airfoil shapes. A good reference is Theory of Wing Sections by Abbot and Von Doenhoff. Data for the four-digit series was published by NACA in the 1920s and 1930s and is still widely used to verify computer code and experimental results. We’ll use NACA four-digit graphs in Sec. 1.3.5 to demonstrate how to read and interpret airfoil data. Appendix C presents data for selected NACA airfoils. 1.3.4.2 Lift, drag, and moment coefficients. Consider an airfoil, at some angle of attack, as shown in Fig. 1.35. Again, lift and drag are the components of the aerodynamic force perpendicular and parallel to the freestream velocity, respectively. In general, the pressure and shear stress distributions also cause a moment (M ), where pitch up (as shown) is considered positive. When comparing airfoil (or for that matter, aircraft) performance, values of lift, drag, and moments are somewhat meaningless. For example, one aircraft might generate twice the lift, but do so very inefficiently, in terms of its design and airspeed. Therefore, dimensionless coefficients are used. Lift, drag, and moment coef- ficients lend themselves beautifully when comparing aerodynamic performance. Their definition and significance stem from a principle called dynamic similar- ity. Consider the flow over two bodies. By definition, the flows are dynamically similar if 1) Geometric similarity exists (the bodies look alike, scale models) and 2) Similarity parameters are the same. If the flows are dynamically similar, then the force and moment coefficients will be equal and the streamline pattern over each body will be geometrically similar. The key is determining the governing similarity parameters. Dimensional analysis (for example, the Buckingham Pi theorem) provides a mechanism. By applying dimensional analysis to an aircraft, 8 the following force=moment coef- ficients are defined: C L ¼ L qq1S ð1:15Þ CD ¼ D qq1S ð1:16Þ CM ¼ M qq1S cc ð1:17Þ Fig. 1.35 Aerodynamic forces and moments on an airfoil. A REVIEW OF BASIC AERODYNAMICS 31 S is a reference area—planform (top view) area of the aircraft’s wing, qq1 is the freestream dynamic pressure ( 1 2 rV 2 1), and cc is the aircraft’s mean aero- dynamic chord (MAC) as defined in Sec. 1.4.1. When discussing airfoils, these forms of the coefficients are inconvenient. Recall from Sec. 1.3, wing tips are considered to approach infinity, thus making the planform area meaningless. To avoid this, airfoil data are presented in terms of lift, drag, and moment, per unit span. Refer to the airfoil section in Fig. 1.36. The distance between wing tips is called the span (b). When collecting airfoil data, this is the width of the wind tunnel’s test section, ensuring that wing tip effects are not included in the force and moment results. Because airfoil sections are not tapered, the mean chord is just the airfoil’s chord. Therefore, the planform area is simply S ¼ b  c Substituting the above for S, and manipulating the equations, leads to the following form of the coefficients. Note that the numerator is now lift per unit span, for example. A lower case is also used to denote airfoil (not aircraft) data. C l ¼ L=b qq1c ð1:18Þ C d ¼ D=b qq1c ð1:19Þ Cm ¼ M =b qq1c2 ð1:20Þ These coefficients, regardless of which form, are a function of three similarity parameters: Mach number, Reynolds number, and angle of attack. Therefore, if a scale model is tested in a wind tunnel, with Reynolds number, Mach number, and angle of attack equal to those in a flight test, the coefficients should accu- rately predict the forces and moments in flight, which is an extremely powerful experimental tool! Fig. 1.36 Airfoil characteristics. 32 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS How do lift, drag, and moment coefficients change with these similarity para- meters? The answer is discussed in later sections. 1.3.4.3 Center of pressure and aerodynamic center. Any point can be chosen on an airfoil to represent the aerodynamic forces (lift and drag) and moment. Before defining two special points, the center of pressure and the aerodynamic center, we will discuss how a moment arises. Figure 1.37 shows the pressure distribution about an airfoil (ignoring shear stress contributions). F1 is the net downward force because of the pressure distribution on the upper surface of the airfoil. Likewise, F2 is the net upward force. If we choose to support the airfoil about an arbitrary point at the quarter chord location (indicated by the black dot), an aerodynamic moment also results about this point. In this case, the airfoil will tend to pitch down about the quarter chord point because of the aerodynamic pressure distribution. For a given angle of attack and Reynolds number, there is one location, called the center of pressure where the aerodynamic moment about that point is zero. Refer to Fig. 1.38. The center of pressure is not a very convenient reference point, in that a change in either angle of attack or Reynolds number (visualize as a change in freestream velocity) will cause the center of pressure location to shift. In contrast, there is one point on the airfoil, called the aerodynamic center, where the moment coefficient about that point remains constant with changes Fig. 1.37 Pressure distribution about an airfoil. Fig. 1.38 Illustration of center of pressure. A REVIEW OF BASIC AERODYNAMICS 33 in angle of attack and Reynolds number. It is fixed on the airfoil and located at approximately the quarter chord for subsonic flow. If you supported an airfoil at the aerodynamic center, the aerodynamic moment about the aerodynamic center would stay constant as angle of attack was changed (and velocity was held constant). Likewise, if both angle of attack and velocity were changed, the aerodynamic moment coefficient about the aerodynamic center would stay constant. See Fig. 1.39. Again, pitch up is positive by our convention. For an airfoil with positive camber in subsonic flow, the moment about the aerodynamic center will be negative (nose down). In transonic and supersonic flow conditions, the location of the aerodynamic center moves aft. 1.3.5 Airfoil Data Recall that airfoil force and moment coefficients are a function of angle of attack, Reynolds number, and Mach number. In this section, we will focus on how angle of attack and Reynolds number affect these coefficients. Mach effects will be discussed in the next section. A typical lift curve (graph of lift coefficient vs angle of attack), for a posi- tively cambered airfoil, is presented in Fig. 1.40. A plot of Cl vs a is one of the classics of aerodynamics. A couple key points about a lift curve follow: 1) At some angle of attack, astall , lift dramatically decreases and the airfoil stalls because of flow separation. 2) Just before stalling, the airfoil reaches a maximum lift coefficient; this is denoted by C lmax . Fig. 1.39 Illustration of aerodynamic center for subsonic flow. 34 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 3) The lift curve slope (denoted by C la ) is typically linear below stall; its slope is about 0.11=deg. 4) At some angle of attack, the lift coefficient is zero; this is called the zero lift angle of attack, or aL¼0. This occurs at a negative angle of attack (chord line oriented below the freestream velocity) for an airfoil with positive camber. At aL¼0 , no lift is generated. 5) Lift is generated when an airfoil with positive camber is at zero degrees angle of attack. This is usually desirable in aircraft design. 6) An increase in Reynolds number tends to increase max lift and delay the onset of stall. Does this make sense? (Hint: think about the physics of stall.) The lift curve for a symmetric airfoil looks basically the same. However, there is one significant difference: the curve passes through the origin. Con- vince yourself that this makes sense. The equation for predicting Cl in the linear region is Cl ¼ Cla ða  aL¼0Þ ð1:21Þ This equation is used to predict C l as a function of angle of attack when Cla and aL¼0 are known. A typical drag polar, or graph of drag coefficient vs lift coefficient, is shown in Fig. 1.41. Because lift coefficient and angle of attack vary linearly (before stall), it is easier to interpret these graphs if lift coefficient is visualized as an angle of attack. High Cl implies high a. Cd is the airfoil’s profile drag coefficient—it includes skin friction and pres- sure drag. Like the lift curve, it is critical to understand what a drag polar is ‘‘saying.’’ Key points: Fig. 1.40 Airfoil lift curve. A REVIEW OF BASIC AERODYNAMICS 35 1) When C d is plotted against C l , the graph is parabolic in nature. As lift increases, drag increases in a parabolic fashion. To explain this, consider what the oncoming flow ‘‘sees’’ at low and high lift coefficients (or a’s). a) Drag is a minimum at the smaller lift coefficients (small a). Here, the oncoming flow ‘‘sees’’ a slender body—skin friction dominates and there is very little pressure drag. b) Drag reaches a maximum at the larger lift coefficients (high a). The airfoil is no longer slender—it is a blunt body! Skin friction drag is still present, but pressure drag becomes increasingly important! 2) This drag polar is for a symmetric airfoil. The data are symmetric about the y axis, with minimum drag at a lift coefficient equal to zero (a ¼ 0). For a positively cambered airfoil, the curve shifts to the right. 3) As with the lift curve, Reynolds number has little effect at the low lift coefficients. However, as the angle of attack increases, Re becomes important. Why? Fig. 1.41 Airfoil drag polar. Fig. 1.42 Moment coefficient about the aerodynamic center lift coefficient. 36 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS Figure 1.42 is a graph of the moment coefficient about the aerodynamic center vs lift coefficient (or angle of attack) for a positively cambered airfoil. The key points are 1) The moment coefficient about the aerodynamic center remains constant with Reynolds number and angle of attack variation. 2) The moment coefficient about the aerodynamic center for an airfoil with positive camber is negative—the airfoil has a pitch-down tendency. Later, we will see that this has important ramifications in terms of longitudinal stability. Example 1.10 A NACA 4412 airfoil with a 2-ft chord and a 5-ft span is being tested in a wind tunnel at standard sea-level conditions and a test section velocity of 240 ft=sec and an angle of attack of 8 deg. What is the airfoil’s maximum thick- ness, maximum camber, location of maximum camber, and zero-lift angle of attack? Also, calculate the lift, drag, and pitching moment about the aero- dynamic center. The airfoil maximum thickness, camber, and location of maximum camber depend only on the NACA 4412 airfoil shape and length of the airfoil chord. The first digit of the 4412 designation specifies a maximum camber, which is 4% of the 2-ft chord or 0.08 ft. The second digit indicates the chordwise location of the point of maximum camber which is 0.4 c or 0.8 ft aft of the leading edge. The last two digits specify a 12% thick airfoil, and there- fore the maximum thickness is 0.12 c or 0.24 ft. The aerodynamic properties of the airfoil may depend on Reynolds number, which for the given test condi- tions is Re ¼ rVc m ¼ ð0:00238Þð240Þð2Þ 3:737  107 ¼ 3:06  10 6 We will thus use the airfoil curves for Re ¼ 3  10 6 . The value of the zero lift angle of attack does not, in fact, vary significantly with Reynolds number as we check the first NACA chart. The Cl at an angle of attack of 8 deg does show some slight variation with Reynolds number. These values are obtained from the NACA 4412 airfoil charts7 as aL¼0 ¼ 4 Cla¼8 ¼ 1:2 A REVIEW OF BASIC AERODYNAMICS 37 The profile drag coefficient and pitching moment coefficient about the aero- dynamic center are obtained from the second chart just shown for a Cl of 1.2 and a Reynolds number of 3  106 . Cd ¼ 0:013 and Cm AC ¼ 0:1 We can then determine the lift, drag, and moment about the aerodynamic center given that S is 10 ft2 (2 ft chord  5 ft span). L ¼ Cl 1 2 rV 2   S ¼ 1:2 1 2 ð0:00238Þð240Þ2   ð10Þ ¼ 822:5 lb D ¼ Cd 1 2 rV 2   S ¼ 0:013 1 2 ð0:00238Þð240Þ2   ð10Þ ¼ 8:91 lb M AC ¼ Cm AC 1 2 rV 2   Sc ¼ 0:1 1 2 ð0:00238Þð240Þ2   ð10Þð2Þ ¼ 137:1 ft  lb Note that the second chart also gives the exact location of the aerodynamic center which is very close to the quarter chord, as previously discussed. 1.3.6 Compressibility (Mach) Effects Earlier we said that lift, drag, and moment coefficients were a function of Reynolds number, angle of attack, and Mach number. The airfoil data we 38 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS examined in the previous section were for low speed or incompressible flow. The influence of Reynolds number and angle of attack were shown, but no compressibility (or Mach) effects were shown. At Mach 0.3, our threshold for compressible flow, the density of air changes approximately 5% from its static value. As Mach number increases beyond 0.3, it is no longer reasonable to ignore the effects of compressibility. Depend- ing on the shape of the body, at Mach numbers approaching the speed of sound and beyond, shock waves develop, significantly changing the aerody- namic properties of the flowfield. Figure 1.43 identifies how the flowfield prop- erties change across a normal shock. In this figure, station 1 is ahead of the normal shock, and station 2 is behind. A shock wave is very thin (on the order of 105 cm) and very viscous.9 Velocity and Mach number abruptly decrease across a shock. Total pressure, which is a measure of the flow’s energy, decreases. All the static properties increase, including pressure. It is this ‘‘shock jump’’ in pressure that has the most profound effect on the force=moment coefficients. To understand how compressibility influences force and moment coeffi- cients, we need to introduce the definition of the critical Mach number (Mcrit ). Consider an airfoil as shown in Fig. 1.44 and assume the freestream Mach number is gradually increased. As the Mach number increases, the properties in the flowfield surrounding the airfoil will naturally change. At some freestream Mach number, called the critical Mach number, sonic flow will first be achieved at a point in the flow- field (usually close to the surface of the airfoil). Figure 1.45 is a qualitative sketch showing the variation of an airfoil’s lift coefficient with Mach number. As you might expect from the rapid changes in lift coefficient, the flowfield is changing dramatically as the Mach number is increased. Figure 1.46, based on flow visualization, shows changes in shock wave formation for the points labeled a through e in Fig. 1.45. The following are the significant points from Figs. 1.45 and 1.46: Fig. 1.43 Property changes across a normal shock. A REVIEW OF BASIC AERODYNAMICS 39 1) The flowfield is subsonic until point a. Typically, a compressibility correction known as the Prandtl–Glauert rule is used in this region. The equation is shown: Cl ¼ C l0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  M 2 p ð1:22Þ Cl0 is an incompressible lift coefficient (the subscript 0 signifies Mach ¼ 0.0). This is the lift coefficient found in typical airfoil data as discussed in Sec. 1.3.5. The Prandtl–Glauert rule is a reasonable correc- tion below the critical Mach number, Mcrit . A rule of thumb is to only use Prandtl–Glauert to Mach 0.7. 2) At point b, the flow is supersonic over most of the upper surface, terminating in a shock wave. Pressure increases across a shock, thus causing increased likelihood of the flow separating from the airfoil’s surface. An adverse pressure gradient is created by the presence of the shock wave. Fig. 1.44 Illustration of critical Mach number. Fig. 1.45 Variation of airfoil lift coefficient with Mach number. 40 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 3) At point c, the flow over the lower surface is essentially all supersonic. The pressure on the lower surface is less than it was at condition b. The upper surface is relatively unchanged. The net effect is a smaller lift coefficient. 4) Between points c and d, the shock waves, on both surfaces, move aft and the lift coefficient increases. 5) As the freestream Mach number increases, a bow shock (or bow wave) forms. Because the velocity decreases across this initial shock, the shocks at the trailing edge are relatively weaker. The pressure differential, between the upper and lower surface, decreases and lift coefficient decreases. It is important to recognize that the dynamic pressure, qq1 ð1=2r1V 2 1) is increasing with Mach number. Therefore, although the lift coefficient may be decreasing, lift can actually be increasing through the dynamic pressure term. L ¼ Cl qq1S At high Mach numbers, the lift coefficient is typically an order of magnitude less than its value at low speeds. A qualitative sketch of how an airfoil’s drag coefficient changes with Mach number is shown in Fig. 1.47. Note the increase in drag preceding Mach 1.0. This is where the term drag barrier initially came from. At some freestream Mach number, beyond Mcrit , drag increases rapidly. This is called the drag divergence Mach number, M DD. Drag divergence is primarily because of the formation of shock waves on the airfoil’s surface. This, in turn, causes drag because of flowfield separation. A typical definition of where the drag divergence Mach number occurs is @CD=@M > 0:1 (Ref. 10). High speed flow and the accompanying compressibility have introduced a new form of drag called wave drag, D wðCd w in coefficient form). This form of drag is only present at transonic and supersonic speeds. In addition to the drag associated with shock-induced flow separation, drag is created simply by the pressure increase across shocks. For example, consider the supersonic flow over a wedge, as shown in Fig. 1.48. Fig. 1.46 Airfoil shock wave formation. A REVIEW OF BASIC AERODYNAMICS 41 Because the pressure behind the oblique shock wave is higher than the free- stream pressure ( p1), an adverse pressure gradient exists that can result in flow separation and additive drag (wave drag). In summary, the airfoil drag coefficient now consists of three contributors: skin friction drag (C d f ), pressure drag (C dp ), and wave drag (Cd w ). C d ¼ C df þ Cd p þ C dw ð1:23Þ Finally, what happens to the moment about the aerodynamic center as Mach number increases? As you might expect, the coefficient will typically change in the transonic region. The most important Mach effect, however, is the fact that the location of the aerodynamic center shifts from roughly the quarter- chord to the mid-chord as supersonic Mach numbers are achieved. As we’ll see later, this shift has a tremendous effect on pitch stability. 1.4 Finite Wings To this point, we have only addressed airfoils, or infinite span wings. Before we attack a complete airplane, this section introduces the aerodynamics associated with wing tips and finite span wings. First, however, we will define some terms used to describe a wing’s geometry. Fig. 1.47 Variation of airfoil drag coefficient with Mach number. Fig. 1.48 Supersonic flow over a wedge. 42 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS 1.4.1 Finite Wing Geometry In our discussion of force and moment coefficients we introduced wing area (S) and span (b). Figure 1.49 (top view of a finite wing) shows these again. The following are some other useful definitions: 1) A wing’s aspect ratio (AR) is defined in Eq. (1.24). It is dimensionless. AR ¼ b2=S ð1:24Þ 2) The root chord (cr) and the tip chord (c t ) are the distances of the chord line at the root and tip, respectively. The ratio of the tip to root chord is called the taper ratio, l. l ¼ c t c r ð1:25Þ A rectangular wing has no taper; a delta wing approaches a taper ratio of zero. 3) A wing’s sweep angle is often defined at the leading edge (LLE ), the quarter-chord (Lc=4 ), or the mid-chord (Lc=2 ). 4) A wing’s mean aerodynamic chord (MAC), or cc, is defined as cc ¼ MAC ¼ 2 S ðb=2 0 c2dy where y is as shown in Fig. 1.49 and c is the chord at any y position. The MAC can be interpreted as the representative chord length for the forces and moments acting on a wing. For a straight, tapered wing, the MAC can be shown to be cc ¼ 2 3 c r l2 þ l þ 1 l þ 1 ! Fig. 1.49 Finite wing geometry. A REVIEW OF BASIC AERODYNAMICS 43 1.4.2 Induced Drag Induced drag is the penalty paid for generating lift on a finite wing. Consider a finite wing as shown on the T-38 aircraft in Fig. 1.50. A wing generates lift by creating a pressure differential between the upper and lower surfaces. Wing tip vortices are generated as the high-pressure air on the lower wing surface near the wing tip seeks the relatively lower pressure on the upper surface. These small ‘‘tornadoes’’ induce a downward component of velocity, called downwash (w). The freestream velocity is displaced through the induced angle of attack (ai), as shown in Fig. 1.51. The wing ‘‘sees’’ Vlocal . Figure 1.52 shows a wing’s cross section. L0 is the component of the aerodynamic force perpendicular to the local velocity. In effect, the lift vector has been rotated aft—a new form of drag, induced drag (Di), is introduced. Fig. 1.50 Generation of wing tip vorticies and downwash. Fig. 1.52 Induced drag description. Fig. 1.51 Induced angle of attack. 44 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS The wing effectively ‘‘sees’’ a lower angle of attack, aeff . Therefore, less lift is generated. Deriving an equation for induced drag is relatively easy (for example, see Ref. 11). The result is presented (Note: A D subscript is used to distinguish this from airfoil drag): D i ¼ C Di qq1S ð1:26Þ where CD i ¼ C2 L peAR ð1:27Þ A few important points about [Eqs. (1.26) and (1.27)] 1) The span efficiency factor is e. A value of 1.0 is optimum and applies for the case of constant downwash along the wing’s span (obtained from an elliptical lift distribution). For other planforms the efficiency is less, typically ranging between 0.95 and 1.0. 2) A high aspect ratio wing reduces induced drag. For this reason, high aspect ratio wings are used on the U-2 reconnaissance aircraft and gliders. 3) Induced drag is proportional to the lift coefficient, squared. Therefore, at high angles of attack (high lift), induced drag dominates. High lift implies greater pressure differentials, thus stronger vortices, and thus more induced drag. 1.4.3 Drag Summary Finite wing geometry (wing tips) introduces a fourth form of drag—induced drag. Skin friction drag, pressure drag, and wave drag (if above the drag diver- gence Mach number) still exist. In summary, the drag coefficient for a finite wing is written as: C D ¼ C d þ C2 L peAR ð1:28Þ where C d ¼ Cd f þ C dp |fflfflfflfflfflffl{zfflfflfflfflfflffl} profile drag þCd w This information is typically presented graphically as a drag polar, as shown in Fig. 1.53 (again, capital subscripts distinguish this from airfoil data). Note the effect of decreasing aspect ratio. 1.4.4 Lift Coefficient for a Finite Wing We modified airfoil drag (C d ) data to account for the effect of wing tips. Similarly, this section describes how airfoil data are used to predict the lift coefficient for a finite wing. Qualitatively, the effect of aspect ratio on the lift curve slope is shown in Fig. 1.54, where CL is used to denote a finite wing lift coefficient. A REVIEW OF BASIC AERODYNAMICS 45 Note the following 1) As aspect ratio decreases, the lift-curve slope (CLa ) decreases. For the same angle of attack, the lift coefficient is smaller for the finite wing. 2) The zero lift angle of attack (aL¼0 ) does not change. Because the wing is not generating lift, wing-tip vortices are not formed. In effect, the finite and infinite cases behave the same. 3) Given an angle of attack, the lift coefficient for the finite wing can be calculated from the following equation, which takes into account a reduced lift-curve slope: CL ¼ CLa ða  aL¼0Þ ð1:29Þ Fig. 1.53 Effect of aspect ratio on the drag polar. Fig. 1.54 Lift curves for infinite and finite wings. 46 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS where CLa ¼ C la 1 þ 57:3Cla peAR ð1:30Þ Again, a capital L (CL) distinguishes this lift coefficient from that of an airfoil. As mentioned, the zero lift angle of attack can be obtained from airfoil data. As we have seen, the lift curve slope for an airfoil (C la ) is approximately 0.11=deg. Unless otherwise specified, this is a reasonable number to use in Eq. (1.30). Example 1.11 An unswept flying wing has an aspect ratio of 10 and incorporates a NACA 4412 airfoil (as in Example 1.10). For a Reynolds number of 6  10 6 and a span efficiency factor of 0.95, find C L and C D at an angle of attack of 4 deg. Using the NACA 4412 airfoil charts in Example 1.10, we find Cl ¼ 0:85 and aL¼0 ¼ 4 deg for the stated angle of attack and Reynolds number. The airfoil drag coefficient is C d ¼ 0:0065 We next find C La for the finite wing using Eq. (1.30) and a Cla ¼ 0:11=deg. CLa ¼ Cla 1 þ 57:3Cla peAR ¼ 0:11 1 þ 57:3ð0:11Þ ð3:14Þð0:95Þð10Þ ¼ 0:0908=deg Equation (1.29) is used to determine C L. CL ¼ C La ða  aL¼0Þ ¼ 0:0908½4 deg  ð4 degފ ¼ 0:7264 CD is determined from Eq. (1.28). C D ¼ C d þ C2 L peAR ¼ 0:0065 þ ð0:7264Þ2 ð3:14Þð0:95Þð10Þ ¼ 0:0242 Notice that the lift coefficient decreases for a finite wing and that the drag coefficient increases. 1.5 Aircraft Aerodynamics We have discussed the aerodynamics of airfoils and finite wings. It is now time to use this essential background to introduce aircraft aerodynamics. Here A REVIEW OF BASIC AERODYNAMICS 47 we will be discussing the aerodynamics of the entire aircraft, including the wing, tail surfaces, and fuselage. 1.5.1 Load Factor, Aerodynamic Coefficients, and Stall Airspeed To begin, we will define load factor (n), or cockpit g where n is the ratio of an aircraft’s lift to its weight, or n ¼ L=W ð1:31Þ For example: at 2 g, an aircraft is generating an amount of lift equal to twice its weight. The lift and drag coefficients for a complete aircraft are defined below, where lift is replaced by nW to keep the equation in its more general form. CL ¼ L qqS ¼ nW qqS C D ¼ D qqS ð1:32Þ Lift and drag include the contributions from not only the wing, but also the fuselage, horizontal=vertical tail, strakes, and external stores. The reference area, S, now typically includes a portion of the fuselage as shown in Fig. 1.55. The slowest speed an airplane can fly in straight, level, and unaccelerated flight is called the stall speed (Vstall ). For these flight conditions, lift is equal to weight (n ¼ 1). The equation for stall speed is derived as follows: L ¼ W ¼ C L qqS ¼ C L 1 2 rV 2S Solving for velocity V ¼ ffiffiffiffiffiffiffiffiffiffiffi 2W rSC L s ð1:33Þ Fig. 1.55 Illustration of aircraft wing reference area. 48 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS At stall, C L becomes equal to CLmax . Vstall ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2W rSC Lmax s ð1:34Þ CLmax is the maximum lift coefficient for the aircraft’s configuration. In the above form, the stall speed is a ‘‘true airspeed.’’ As altitude increases, density decreases, and an aircraft stalls at a higher true airspeed—not very convenient in terms of flight operations. In Chapter 3, we’ll introduce other airspeeds (indicated, calibrated, etc.) and see how to avoid this inconvenience. Example 1.12 Determine the stall airspeed at sea level for a 10,000-lb T-38 with 20-deg flaps. The wing reference area is 170 ft 2 . Also, determine the load factor if the same aircraft is at an angle of attack of 10 deg with the flaps up at sea level with an airspeed of 265 kn. Use the following chart. (Source: Department of Aeronautics, USAF Academy) A REVIEW OF BASIC AERODYNAMICS 49 To determine the stall speed, we first determine C Lmax for 20-deg flaps from the chart. CLmax ¼ 0:88 Then, Eq. (1.34) can be used to find the stall speed. Vstall ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2W rSC Lmax s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð10;000Þ ð0:00238Þð170Þð0:88Þ s ¼ 237 ft=s To find the load factor, the lift coefficient for an angle of attack of 10 deg (flaps up) is found from the chart. CL ¼ 0:67 Then, Eq. (1.32) is solved for load factor. n ¼ CL qqS W ¼ ð0:67Þ 1 2 ð0:00238Þ 265  1:69 ft=s kn  2 ! ð170Þ 10;000 ¼ 2:72 At these conditions, the T-38 is pulling 2.72 g. 1.5.2 Aircraft Drag Polar Typically, the aerodynamics of an airplane are presented as a drag polar— this is in the form of an equation, a graph, or both. Recall from the airfoil and finite wing discussions that a drag polar (by definition) shows the relationship between lift and drag coefficients for a specific aerodynamic body. In equation form, the drag polar of an aircraft is CD ¼ CD0 þ C2 L peAR ð1:35Þ or simply CD ¼ CD0 þ KC2 L ð1:36Þ CD0 is called the parasite drag coefficient or zero lift drag coefficient. Below the drag divergence Mach number, it is approximated by a constant (indepen- dent of lift) for a specific aircraft configuration. Included in this term are profile drag (skin friction and zero lift pressure drag) and interference drag. Interference drag is generated when more than one body (for example, stores on a wing) is placed in the same flowfield, creating eddies, turbulence, and=or restrictions to smooth flow. For example, if an external store is hung on a wing, the combined drag will typically be more than the summation of the 50 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS individual store and wing drag. Occasionally, the addition of blended surfaces, as in the case of the F-15 conformal fuel tanks, will reduce the interference drag. The term C2 L=peAR in Eq. (1.35) is the drag due to lift, or induced drag term and is sometimes referred to as C Di . The induced drag on all lifting surfaces (wing, strakes, and horizontal tail) and the increment of pressure drag when the aircraft is generating lift are included in this term. The term K in Eq. (1.36) is referred to as the induced drag factor and can be seen to be equal to 1=peAR. If Mach effects are unimportant, K is a constant for a specific config- uration. The term e is called the Oswald efficiency factor. It can be thought of as a ‘‘fudge factor,’’ obtained through wind-tunnel and=or flight tests, which takes into account such effects as a nonelliptical lift distribution and the varia- tion of pressure drag with lift. Typically, e is on the order of 0.8, but no greater than 1.0. It is very convenient to present the drag polar graphically. Typically, this is done in two ways, as presented in Fig. 1.56. We have previously seen the parabolic presention of the drag polar. The second graph is a linear presentation because C D is plotted as a function of C2 L. This form of the drag polar is convenient for determining the induced drag factor, K, which is simply the slope of the line. It is also a convenient format for plotting individual flight test data points when determination of the drag polar is the end objective. In this format, a linear curve fit to the data is supported by theory. Mach effects are typically defined through the values of C D0 and K. For example, Table 1.2 illustrates how these ‘‘constants’’ change for the F-16. Another useful measure of drag used by aircrews is the drag count. A drag count is defined as one ten thousandth of a drag coefficient, or 1 drag count ) a C D of 0:0001 The drag count is simply a ‘‘user friendly’’ way to express the drag coefficient. A C D of 0.025 is equivalent to 250 drag counts. Drag counts are especially useful when adding external protuberances to an aircraft. For example, the addition of a forward radome on the AC-130H gunship adds approximately 23 drag counts to the total aircraft drag. This is equivalent to a drag coefficient Fig. 1.56 Two ways of presenting the drag polar. A REVIEW OF BASIC AERODYNAMICS 51 increase of 0.0023, but 23 drag counts proves to be an easier number to remember. Example 1.13 Using Table 1.2, find the total drag counts for the F-16 at a lift coefficient of 0.2 and Mach 0.86. From Table 1.2, we have, CD0 ¼ 0:0169 and K ¼ 0:117 Using Eq. (1.36), C D ¼ CD0 þ KC2 L ¼ 0:0169 þ ð0:117Þð0:2Þ2 ¼ 0:02158 For this condition, the aircraft would have 215.8 drag counts. 1.5.3 Total Aircraft Drag The total drag on an aircraft is simply D ¼ CD qqS It is useful to factor out velocity (V1) in the previous equation, as shown: D ¼ ½C D0 þ KC2 LŠ 1 2 rV 2S Substituting in C L ¼ nW qqS ¼ nW 1 2 rV 2S Table 1.2 Variation of C D0 and K with Mach number for the F-16 (Ref. 2) Mach C D0 K 0.1 0.0169 0.117 0.86 0.0169 0.117 1.05 0.0430 0.128 1.5 0.0382 0.252 2.0 0.0358 0.367 52 INTRODUCTION TO AIRCRAFT FLIGHT MECHANICS We have D ¼ C D0 1 2 rS   parasite drag V 2 þ 2KðnW Þ2 rS " # induce