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MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS. Overview of Compressible Flows: Critical Mach Number and Wing Sweep April 25, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk. EXAMPLES: INFLUENCE OF COMPRESSIBILITY. M ∞ < 1. M ∞ ~ 0.85.

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MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

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  1. MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS Overview of Compressible Flows: Critical Mach Number and Wing Sweep April 25, 2011 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

  2. EXAMPLES: INFLUENCE OF COMPRESSIBILITY M∞ < 1 M∞ ~ 0.85 M∞ > 1

  3. WHEN IS FLOW COMPRESSIBLE?

  4. WHEN IS FLOW COMPRESSIBLE?

  5. EXAMPLES: COMPRESSIBLE INTERNAL FLOW

  6. EXAMPLE: H2 VARIABLE SPECIFIC HEAT, CP

  7. COMPRESSIBILITY SENSITIVITY WITH g

  8. PRESSURE COEFFICIENT, CP • Use non-dimensional description instead of plotting actual values of pressure • Pressure distribution in aerodynamic literature often given as Cp • So why do we care? • Distribution of Cp leads to value of cl • Easy to get pressure data in wind tunnel • Shows effect of M∞ on cl

  9. EXAMPLE: CP CALCULATION See §4.10

  10. COMPRESSIBILITY CORRECTION:EFFECT OF M∞ ON CP Cp at a point on an airfoil of fixed shape and fixed angle of attack For M∞ < 0.3, r ~ const Cp = Cp,0 = 0.5 = const Flight Mach Number, M∞

  11. COMPRESSIBILITY CORRECTION:EFFECT OF M∞ ON CP Effect of compressibility (M∞ > 0.3) is to increase absolute magnitude of Cp as M∞ increases Called: Prandtl-Glauert Rule For M∞ < 0.3, r ~ const Cp = Cp,0 = 0.5 = const M∞ Prandtl-Glauert rule applies for 0.3 < M∞ < 0.7

  12. EXAMPLE: SUPERSONIC WAVE DRAG F-104 Starfighter

  13. CRITICAL MACH NUMBER, MCR • As air expands around top surface near leading edge, velocity and M will increase • Local M > M∞ Flow over airfoil may have sonic regions even though freestream M∞ < 1 INCREASED DRAG!

  14. CRITICAL FLOW AND SHOCK WAVES MCR • Sharp increase in cd is combined effect of shock waves and flow separation • Freestream Mach number at which cd begins to increase rapidly called Drag-Divergence Mach number

  15. CRITICAL FLOW AND SHOCK WAVES ‘bubble’ of supersonic flow

  16. CRITICAL FLOW AND SHOCK WAVES MCR

  17. EXAMPLE: IMPACT ON AIRFOIL / WING DRAG Only at transonic and supersonic speeds Dwave= 0 for subsonic speeds below Mdrag-divergence Profile Drag Profile Drag coefficient relatively constant with M∞ at subsonic speeds

  18. AIRFOIL THICKNESS SUMMARY • Which creates most lift? • Thicker airfoil • Which has higher critical Mach number? • Thinner airfoil • Which is better? • Application dependent! Note: thickness is relative to chord in all cases Ex. NACA 0012 → 12 %

  19. CAN WE PREDICT MCR? A • Pressure coefficient defined in terms of Mach number (instead of velocity) PROVE THIS FOR CONCEPT QUIZ • In an isentropic flow total pressure, p0, is constant • May be related to freestream pressure, p∞, and static pressure at A, pA

  20. CAN WE PREDICT MCR? • Combined result • Relates local value of CP to local Mach number • Can think of this as compressible flow version of Bernoulli’s equation • Set MA = 1 (onset of supersonic flow) • Relates CP,CR to MCR

  21. HOW DO WE USE THIS? • Plot curve of CP,CR vs. M∞ • Obtain incompressible value of CP at minimum pressure point on given airfoil • Use any compressibility correction (such as P-G) and plot CP vs. M∞ • Intersection of these two curves represents point corresponding to sonic flow at minimum pressure location on airfoil • Value of M∞ at this intersection is MCR 1 3 2

  22. IMPLICATIONS: AIRFOIL THICKNESS • Thick airfoils have a lower critical Mach number than thin airfoils • Desirable to have MCR as high as possible • Implication for design → high speed wings usually design with thin airfoils • Supercritical airfoil is somewhat thicker Note: thickness is relative to chord in all cases Ex. NACA 0012 → 12 %

  23. THICKNESS-TO-CHORD RATIO TRENDS A-10 Root: NACA 6716 TIP: NACA 6713 Thickness to chord ratio, % F-15 Root: NACA 64A(.055)5.9 TIP: NACA 64A203 Flight Mach Number, M∞

  24. ROOT TO TIP AIRFOIL THICKNESS TRENDS Boeing 737 Root Mid-Span Tip http://www.nasg.com/afdb/list-airfoil-e.phtml

  25. SWEPT WINGS • All modern high-speed aircraft have swept wings: WHY?

  26. WHY WING SWEEP? V∞ V∞ Wing sees component of flow normal to leading edge

  27. WHY WING SWEEP? V∞ V∞,n W W V∞ V∞,n<V∞ Wing sees component of flow normal to leading edge

  28. SWEPT WINGS: SUBSONIC FLIGHT • Recall MCR • If M∞ > MCR large increase in drag • Wing sees component of flow normal to leading edge • Can increase M∞ • By sweeping wings of subsonic aircraft, drag divergence is delayed to higher Mach numbers

  29. SWEPT WINGS: SUBSONIC FLIGHT • Alternate Explanation: • Airfoil has same thickness but longer effective chord • Effective airfoil section is thinner • Making airfoil thinner increases critical Mach number • Sweeping wing usually reduces lift for subsonic flight

  30. SWEPT WINGS: SUPERSONIC FLIGHT • If leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic → Large Wave Drag • If leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic → Reduced Wave Drag • For supersonic flight, swept wings reduce wave drag

  31. WING SWEEP COMPARISON F-100D English Lightning

  32. SWEPT WINGS: SUPERSONIC FLIGHT M∞ < 1 SU-27 q M∞ > 1 • ~ 26º m(M=1.2) ~ 56º m(M=2.2) ~ 27º

  33. WING SWEEP DISADVANTAGE • Wing sweep beneficial in that it increases drag-divergences Mach number • Increasing wing sweep reduces the lift coefficient • At M ~ 0.6, severely reduced L/D • Benefit of this design is at M > 1, to sweep wings inside Mach cone

  34. TRANSONIC AREA RULE • Drag created related to change in cross-sectional area of vehicle from nose to tail • Shape itself is not as critical in creation of drag, but rate of change in shape • Wave drag related to 2nd derivative of volume distribution of vehicle

  35. EXAMPLE: YF-102A vs. F-102A

  36. EXAMPLE: YF-102A vs. F-102A

  37. CURRENT EXAMPLES • No longer as relevant today – more powerful engines • F-5 Fighter • Partial upper deck on 747 tapers off cross-sectional area of fuselage, smoothing transition in total cross-sectional area as wing starts adding in • Not as effective as true ‘waisting’ but does yield some benefit. • Full double-decker does not glean this wave drag benefit (no different than any single-deck airliner with a truly constant cross-section through entire cabin area)

  38. EXAMPLE OF SUPERSONIC AIRFOILS http://odin.prohosting.com/~evgenik1/wing.htm

  39. SUPERSONIC AIRFOIL MODELS • Supersonic airfoil modeled as a flat plate • Combination of oblique shock waves and expansion fans acting at leading and trailing edges • R’=(p3-p2)c • L’=(p3-p2)c(cosa) • D’=(p3-p2)c(sina) • Supersonic airfoil modeled as double diamond • Combination of oblique shock waves and expansion fans acting at leading and trailing edge, and at turning corner • D’=(p2-p3)t

  40. APPROXIMATE RELATIONS FOR LIFT AND DRAG COEFFICIENTS

  41. http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/home.htmCASE 1: a=0° Expansion Shock waves

  42. CASE 1: a=0°

  43. CASE 2: a=4° Aerodynamic Force Vector Note large L/D=5.57 at a=4°

  44. CASE 3: a=8°

  45. CASE 5: a=20° At around a=30°, a detached shock begins to form before bottom leading edge

  46. CASE 6: a=30°

  47. DESIGN OF ASYMMETRIC AIRFOILS

  48. QUESTION 9.14 • Consider a diamond-wedge airfoil as shown in Figure 9.36, with half angle e=10° • Airfoil is at an angle of attack a=15° in a Mach 3 flow. • Calculate the lift and wave-drag coefficients for the airfoil. Compare with your solution

  49. EXAMPLE: MEASUREMENT OF AIRSPEED • Pitot tubes are used on aircraft as speedometers (point measurement) Subsonic M < 0.3 Subsonic M > 0.3 Supersonic M > 1 M < 0.3 and M > 0.3: Flows are qualitatively similar but quantitatively different M < 1 and M > 1: Flows are qualitatively and quantitatively different

  50. MEASUREMENT OF AIRSPEED:INCOMPRESSIBLE FLOW (M < 0.3) • May apply Bernoulli Equation with relatively small error since compressibility effects may be neglected • To find velocity all that is needed is pressure sensed by Pitot tube (total or stagnation pressure) and static pressure Comment: What is value of r? • If r is measured in actual air around airplane (difficult to do) • V is called true airspeed, Vtrue • Practically easier to use value at standard seal-level conditions, rs • V is called equivalent airspeed, Ve Static pressure Dynamic pressure Total pressure Incompressible Flow

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