AE317 Aircraft Flight Mechanics & Performance UNIT B: Wings and Airplanes Brandt, et.al., Introduction to Aeronautics: ROAD MAP . A Design Perspective Chapter 4: Wings and Airplanes 4.6 Mach-Number Effects 4.8 More Details B-1: Wings, H-L Devices, & Whole Aircraft B-2: Mach-Number Effects & Lift/Drag B-3: Advanced Subjects & An Exercise Unit B-2: List of Subjects Mach Waves Shock Waves Flight Regimes Area Ruling & Wing Sweep Supersonic Drag Whole Aircraft Lift Calculations Horizontal Stabilizers and Canards Whole Aircraft Drag Calculations Page 2 of 13 Unit B-2 Mach Waves (4.19) Mach Number • Freestream Mach number is defined as: M V a (4.17); where, a= RT (4.18) : Ratio of specific heat, ==ccpv1.4 (for standard air) T : Absolute temperature R : Gas constant, 287 J( kg K) or 1,716 (ft lb) ( slug o R) Mach Waves and Mach Angle • Consider an infinitesimally small body moving in the atmosphere (the body is making small pressure disturbances, or sound waves): sound wave propagation behavior depends on the body's Mach number. • M = 0: the sound waves radiate outward in concentric circles from the body. • M < 1: the sound waves upstream of it are closer together. • M = 1: the body is moving at the same speed as the sound waves it emits, so that all of the sound emitted by the body reaches a point ahead of it at the same time the body does. The sound waves collect into a single pressure wave, known as a Mach wave, which is perpendicular to the direction of movement of the body. • M > 1: the Mach wave trails back from the body at an angle, called the Mach angle: −−11 ==sin(a V) sin( 1 M ) (4.19) Mechanism of Shock Waves • The pressure waves caused by a body moving through the air (aircraft, missile, rocket, etc.), likewise, influence the flow field ahead of the body. • The influence of the high pressure at a stagnation point at the front of the body is transmitted upstream at the speed of sound, so that the flow slows down gradually (rather than suddenly). • If the speed of the body through the air exceeds the speed of sound waves, this process of "warning" the air ahead of the body is approaching becomes impossible. The pressure change occurs suddenly in a short distance (formation of a shock wave). • A shock wave undergoes a rapid rise in pressure, density, and temperature; a rapid decrease in velocity; and a loss of total pressure. • The angle of a shock wave is not the same as the Mach angle. Page 3 of 13 Unit B-2 Shock Waves M < 1 M < 1 Expansion Fan Expansion Fan Oblique Shock Critical Mach Number • As an aircraft airspeed increases, even though the freestream (the moving airspeed of the aircraft itself) is still subsonic, a local flow over the "most accelerated" (typically upper surface) location becomes sonic (M = 1). The freestream Mach number at which the local Mach number first reaches sonic (M = 1) is called the critical Mach number (M crit ) . • At MM = crit , no shock wave forms because the flow is sonic (M = 1) only at a single point. • As M increases above M crit , the region of supersonic (M > 1) begins to grow. Pressure waves from decelerating flow downstream of the supersonic region cannot move upstream into the region (a shockwave, called the "termination shock" will be formed and will cause the flow to separate (called, the "shock-induced separation"). This will cause a significant increase in drag and decrease in lift. Drag-Divergence Mach Number (MMcrit → 1) • As increases further, the sudden sharp rise in drag as approaches to 1 (once this was thought to be an physical barrier, called the sound barrier). The Mach number at which this rapid rise in drag occurs is called the drag-divergence Mach number (M DD ) . Supersonic Flight • If is greater than 1 (supersonic), depending on the body shape, as well as the flight Mach number, a shock wave (either detached bow shock or attached oblique shock) will be formed to slow down the supersonic flow to subsonic flow over an aircraft. Page 4 of 13 Unit B-2 Flight Regimes (4.20) (4.21) Flight Regimes • The range of Mach numbers at which aircraft fly is divided up into flight regimes. The regimes are chosen based on the aerodynamic phenomena that occur at Mach numbers within each regime and on the types of analysis that must be used to predict the consequences of those phenomena. Lift-Coefficient-Curve Slope C Variations ( L ) • Lift-coefficient-curve slope of most aircraft vary with Mach number. C L M =0 • For subsonic, 0.3 MM: C = (4.20) Prandtl-Glauert Correction (Note that for crit L 2 1− M incompressible subsonic, M 0.3 , the correction becomes trivial). 4 57.3 • For supersonic, M 1 cos : C = (per deg) (4.21) LE L 2 M −1 • In the transonic regime, with many complex physical phenomena (i.e., shock-induced separation), it is very difficult to predict the behavior of C . L • For a well-designed supersonic aircraft, levels off from the supersonic curve defined by eq(4.20) and transition smoothly to the supersonic curve defined by eq(4.21). Page 5 of 13 Unit B-2 Area Ruling & Wing Sweep (4.22) Drag at High Subsonic Mach Numbers • Drag is caused by many complex parameters, and high Mach numbers only add more complexity. In subsonic regime, primary changes in parasite (or zero-lift) drag coefficient C and Oswald's ( D0 ) efficiency factor (e0 ) for a given aircraft are caused by increasing Reynolds number as Mach number increases. • However, these changes are often negligible with Mach number below M crit . Supersonic Zero-Lift Drag • For the supersonic flight regime, wave drag (drag due to the formation of shock wave) will be added on top of all other types of drag components. It is well known that at supersonic speeds, slender and pointed bodies with cross-sectional areas varying in certain shape (known as: Sears-Haack body area distribution) have minimum wave drag. The mathematical relationship will generate a "wasp waist" shape of aircraft fuselage, called the area rule. 2 4.5 Amax CD = (4.22) wave Sl Effect of Wing Sweep • Aircraft wing's M crit can be raised by sweeping the wing. Sweeping the wing increases the effective chord length. Increasing the airfoil chord length lowers the airfoil's thickness-to-chord ratio. This in turn reduces the component of the flow speeding up to get past airfoil. Page 6 of 13 Unit B-2 Supersonic Drag Total Drag of an Aircraft at High-Speed • The total drag, C , on an aircraft is the sum of parasite (or "zero-lift") drag, C , and induced drag D D0 (or "drag due-to-lift"), kC 2 , that is: C=+ C kC 2 (4.23b). L DDL0 • Note that zero-lift drag is composed of profile drag (the subsonic drag not caused by lift), C , and Dp wave drag, C , that is: CCC=+ (4.23a). Dwave DDD0p wave Shock Cone of a Supersonic Flight • A further consideration for the shape of supersonic aircraft is the benefit to be gained by keeping the wing inside the shock-wave cone generated by the aircraft's nose. • This practice reduces the aircraft's wave drag because the Mach number inside the cone is lower than the freestream Mach number (M ) , and shock waves are weaker than they would be, if the wing were exposed to the full M . • If a wing tip or other component projects outside the shock cone, it will generate an additional shock wave. This further increases total wave drag on the aircraft and, where the two shock waves intersect or interfere, causes additional disruption to the flow. • In the case of very high Mach numbers, this shock wave interaction contributes to additional heating of the aircraft's skin, which can lead to structural damage. Page 7 of 13 Unit B-2 Example B-2-1 (Mach-Number Effects) Example 4.3 SR-71 was a mystery aircraft in the 1970-80s. Its maximum speed was a carefully kept secret. The practice of designing every part of an aircraft to fit inside the shock cone (generated by its nose) is especially important for SR-71 (high-speed flight). At high-speed, impingement of a shock wave on a wing’s leading edge could cause excessive heating. With a figure of SR-71 (Fig. 4.32), can you predict SR-71’s maximum flight Mach number? Solution (4.3) _________________________________________________________ ____________ _____________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ _________________________________________________________ Page 8 of 13 Unit B-2 Whole Aircraft Lift Calculations (1) Whole Aircraft Lift Calculations • A complete aircraft will frequently generate significantly more lift than its wing alone. An estimate of a whole aircraft's lift can be made by summing the lift contributions of its various components. The method is suitable for in early conceptual phase design of aircraft. Wing Contribution • The majority of the lift is generated by the wing. The empirical expression for span efficiency factor can be given by: 2 e = (4.24) 2−AR + 4 + AR22 1 + tan ( tmax ) : Sweep angle of the line connecting the point of maximum thickness on each airfoil tmax • One effect of airfoil camber and wing twist on lift is to shift the zero-lift angle of attack. A way to avoid the need for predicting zero-lift angle of attack early in the design process is to work in terms of absolute angle of attack (a ) : aL=− =0 (4.25).