MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS LECTURE 42: APRIL 25, 2005

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MAE 3241 AERODYNAMICS AND FLIGHT
MECHANICSLECTURE 42 APRIL 25, 2005

• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk
• Spring 2005

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IMPLICATIONS AIRFOIL THICKNESS
Note thickness is relative to chord in all
cases Ex. NACA 0012 ? 12

• 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

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THICKNESS-TO-CHORD RATIO TRENDS
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DRAG DIVERGENCE MACH NUMBER

• Sharp increase in cd is combined effect of shock
  waves and flow separation
• Freestream Mach number at which cd begins to
  increase rapidly is defined as Drag-Divergence
  Mach number
• Modern airfoils may operate at point b

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SWEPT WINGS

• 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

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SWEPT WINGS

• Recall MCR
• If M8 gt MCR large increase in drag
• Wing sees component of flow normal to leading
  edge
• Can increase M8
• By sweeping wings of subsonic aircraft, drag
  divergence is delayed to higher Mach numbers

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WING SWEEP DISADVANTAGE

• At M 0.6, severely reduced L/D
• Benefit of this design is at M gt 1, to sweep
  wings inside Mach cone

• Wing sweep beneficial in that it increases
  drag-divergences Mach number
• Increasing wing sweep reduces the lift coefficient

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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

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EXAMPLE YF-102A vs. F-102A
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EXAMPLE YF-102A vs. F-102A
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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)

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SUPERCRITICAL AIRFOILS

• Supercritical airfoils designed to delay and
  reduce transonic drag rise, due to both strong
  normal shock and shock-induced boundary layer
  separation
• Relative to conventional, supercritical airfoil
  has reduced amount of camber, increased leading
  edge radius, small surface curvature on suction
  side, and a concavity in rear part of pressure
  side

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SUPERCRITICAL AIRFOILS
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SUPERCRITICAL AIRFOILS

• For given thickness, supercritical airfoil allows
  for higher cruise velocity
• For given cruise velocity, airfoil thickness may
  be larger
• Structural robustness, lighter weight, more
  volume for increased fuel capacity

757 wing
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SMALL PERTURBATION VELOCITY POTENTIAL EQUATION

• Equation is a linear PDE and easy to solve
• Recall
• Equation is no longer exact
• Valid for small perturbations
• Slender bodies
• Small angles of attack
• Subsonic and Supersonic Mach numbers
• Keeping in mind these assumptions equation is
  good approximation
• Subsonic 1 - M82 gt 0 (elliptic)
• Supersonic 1 - M82 lt 0 (hyperbolic)

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SUPERSONIC APPLICATION

• Linearized small perturbation equation
• Re-write for supersonic flow
• Solution has functional relation
• May be any function of (x - ly)
• Perturbation potential is constant along lines of
  x ly constant

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KEY RESULTS SUPERSONIC FLOWS

• Linearized supersonic pressure coefficient
• Expression for lift coefficient
• Thin airfoil or arbitrary shape at small angles
  of attack
• Expression for drag coefficient
• Thin airfoil or arbitrary shape at small angles
  of attack

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EXAMPLE FLAT PLATE