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 Jkg( K or) 1,716 ft( lbslugR) ( o )
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 1cos : 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: CCkC=+2 (4.23b). L DDL 0 • 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 DDD0wave p
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) 241tan−+++ARAR 22 ( 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). Because of the way this is defined, it is always equal to zero when lift is zero. • For take-off and landing, the maximum amount of aircraft's rotation is the maximum usable absolute angle of attack , and this is usually well below the wing's stall angle (tail clearance ( amax ) for take-off, and pilot's view angle for landing). The maximum usable lift coefficient is limited by this. CCC= = − Lmax L a max L( max L= 0 ) (4.26)
Contribution of High-Lift Devices
• Most flaps change but not C , so the effect can be represented as an increment to the L=0 L maximum usable absolute angle of attack. For flaps that do not span the entire wing:
S f = cos (4.27) aa2−D S h.l.
h.l. : Sweep angle of the flap hinge line
S f : Area of that part of the wing that has the flaps attached to it
• After a is estimated, the maximum usable lift coefficient with flaps is approximated: CCC + (4.28) Lmax Lmax (no flap) L a
Page 9 of 13 Unit B-2 Whole Aircraft Lift Calculations (2)
Maximum Trimmable Lift Coefficient
• High-lift devices do not come without cost. They add weight and complexity, and trailing edge flaps also create a strong nose-down pitching moment. This moment must be counteracted or flaps cannot be used. • Counteracting moments such as these so that the net moment on the aircraft is zero is called trimming of the aircraft. • A common way to do this is to provide the aircraft with a trimming surface such as a canard or horizontal tail that can generate lift in such a way that it creates a moment opposite and equal to the moment created by the flaps.
• (NACA 0006 Example) An airfoil with 60-deg split flap experiences: a −12 deg increment in L=0 and a −0.2 increment in C . A reasonable assumption is that the trimming surface can devote mc 4 about 10 degrees of deflection to trimming the flap moment, leaving the rest of its deflection and lifting capability for maneuvering the aircraft:
CxxL −10 trim surfacewing ( trim surface ) cc44Strim surface CM = (4.29) maxtrimming cS • The assumption allows us to write an equation for maximum trimmable lift increment based on specified trimming surface geometry and assumed equality of the magnitude of trimming surface moment and flap moment: L=0 =CCCLLM maxtrimmable wingC max trimming mc 4
CL −10 xtrim surface x wing 12 ( trim surface ) cc44Strim surface =CCLL (4.30) maxtrimmable wing 0.2 cS • Hence, the maximum trimmable lift coefficient for this case (NACA 0006) is:
CCCLLL= + (4.31) maxtrimmable max clean max trimmable
Page 10 of 13 Unit B-2 Horizontal Stabilizers and Canards
Contribution of Fuselage and Strakes
• For fuselages with strakes or leading-edge extensions, the contribution to the lift generation is: SS+ CC= strake (4.32) LL(with strakewithout) strake( ) S
Horizontal Stabilizers and Canards
• It is usually sufficient to treat horizontal stabilizers and canards (i.e., horizontal surfaces) as additional wings. However, the upwash/downwash caused by the main wing will change the effective angle of attack of these devices. • To determine the contribution of these devices' to the whole aircraft's lift, it is first necessary to determine the rate at which upwash/downwash angle ( ) changes with changing aircraft angle of attack () : 0.25 21 C c −L avg 103 zh =−0.5 1 (4.33) Horizontal Tail ARlb h 7
ca v g : mean geometric chord of the wing
lh : distance between quarter-chord points (from the main wing to the horizontal tail)
zh : vertical distance of the horizontal tail above the plane of the main wing • Once is predicted, then the contribution to the lift-coefficient can be determined: S =−CC1 t (4.34) Horizontal Tail L (due to horizontal tail) L tail S • For a horizontal surface, placed in front of the main wing (i.e., canard): −+(1.04 6 AR−1.7 ) uc0.3 l =−(0.3AR 0.33) (4.35) Canard c
u : upwash angle
lc : distance between quarter-chord points (from the main wing to the canard) S =+CC1 uc L (due to canard) L (4.36) Canard canard S
Whole Aircraft Lift-Curve Slope
• In summary, the whole aircraft's lift-curve slope is the summation of all contributions: CCC= + (4.37) Whole Aircraft Lift LLL(whole aircraft) ( wing+body+strake) ( horizontal surfaces)
Page 11 of 13 Unit B-2 Whole Aircraft Drag Calculations (1)
Whole Aircraft Drag Calculations
• For the whole aircraft, drag is identified as either (i) parasite (zero-lift) drag or (ii) induced drag (drag due-to-lift). CCk=++ Ck C 2 keAR=1 DDLL 0 12 (Note) 10( ) (4.38) Whole Aircraft Drag
• To model a situation where minimum drag occurs at a positive value of lift coefficient, k2 is introduced. This will add the effect of shifting the entire drag polar.
Parasite Drag
• Skin-friction drag on a complete aircraft configuration is generally much greater than that on the
wing alone, because the whole aircraft's wetted area (Swet ) is greater than the wing alone. • A good estimate of subsonic parasite drag can be done from drag data for similar aircraft using the concept of an equivalent skin-friction drag coefficient C : ( fe ) S S CC= (4.40) => CC= wet (4.41) Whole Aircraft Drag fDe 0 Df0 e Swet S
Supersonic Wave Drag
• For aircraft with reasonably smooth area distributions, C can be predicted as: Dwave 2 4.5 Amax CEMDC=+−− M WDLE(0.74 0.37cos1 0.3) (4.42) wavemax Sl D0 0.2 Where, M C =1 cos LE (4.43) D0max
EWD : wave drag efficiency factor (average ~ 2.0 for typical supersonic aircraft)
• Critical Mach number (M crit ) is an important design parameter for high-speed aircraft, and airfoil shape needs to be carefully chosen to make it as high as possible. NACA 64-series airfoils curve-fit
provides an empirical estimation of M crit : 0.6 tmax M crit =−1.0 0.065 100 (4.44) c Page 12 of 13 Unit B-2 Whole Aircraft Drag Calculations (2)
Effect of Wing Sweep
• Aircraft wing's critical Mach number (M crit ) can be raised by sweeping:
ttmaxmax cc(swept wing) =( unswept wing) cos LE (4.45) => =cos LE (4.46) cc(swept wingunswept) wing ( ) • Hence, the critical Mach number for swept wing can be estimated by combining with eq(4.44): t 0.6 0.6 max 0.6 M crit=1.0 − 0.065cos LE 100 => MMcritLE=−−1.0cos1.0 crit( unswept) (4.47) c
Fuselage Contribution
• Only fuselages with relatively blunt noses will produce a value for M crit that is lower than the one determined by the shape of the wing.
Drag due to Lift
• Predicting drag due to lift must begin with predicting Oswald's efficiency factor, e0 . This is done with a curve fit of wind tunnel data for a variety of wing and wing-body combinations: 0.53 0.1 eAR0LE=−−4.6( 1 0.033cos3.3)( ) (4.48)
Effect of Camber
• The lift coefficient where minimum drag coefficient for an aircraft occurs is defined as: CLDmin .
This can be represented by: k2=−2 k 1 CLD min (4.49). A crude approximation of a cambered airfoil is that the airfoil generates minimum drag when at zero angle of attack and effect of induced drag is to
move to a value halfway between zero and value of CL when = 0 :
−L=0 CCCCL= L( a) = L( − L==00) = L( − L ) . Thus, CCLL= (4.51) =0 min D 2
Page 13 of 13 Unit B-2 Whole Aircraft Drag Calculations (2) (Continued)
• For aircraft with minimum drag at nonzero lift (cambered airfoil), this leads to the following revised drag polar predictions:
Swet 2 CC= (4.52) => CDDL=+ C k1 C (4.53) Dfmin e S 0 min min D • For supersonic flight, wave drag must be added to the subsonic C to get the total supersonic value D0 of . Because cambered airfoils generate more wave drag in supersonic flight than symmetric airfoil.
Supersonic Drag due to Lift
• For a typical high-speed aircraft, minimum drag at zero angle of attack (means: k2 = 0 ) and: AR( M 2 −1) k1=cos LE (4.54) (4AR M 2 −− 1) 2