MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS

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MAE 3241 AERODYNAMICS ANDFLIGHT MECHANICS

• Design of Supersonic Airfoils
• April 18, 2007
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk

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

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APPROXIMATE RELATIONS FOR LIFT AND DRAG
COEFFICIENTS
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CASE 1 a0
Expansion
Shock waves
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CASE 1 a0
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CASE 2 a4
Aerodynamic Force Vector
Note large L/D5.57 at a4
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CASE 3 a8
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CASE 5 a20
At around a30, a detached shock begins to form
before bottom leading edge
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CASE 6 a30
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DESIGN OF ASYMMETRIC AIRFOILS
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QUESTION 9.14

• Consider a diamond-wedge airfoil such as shown in
  Figure 9.36, which a half angle e10. The
  airfoil is at an angle of attack a15 to a Mach
  3 freestream. Calculate the lift and wave-drag
  coefficients for the airfoil.

Compare with your solution
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EXAMPLE MEASUREMENT OF AIRSPEED

• Pitot tubes are used on aircraft as speedometers
  (point measurement)

Subsonic M lt 0.3
Subsonic M gt 0.3
Supersonic M gt 1
M lt 0.3 and M gt 0.3 Flows are qualitatively simil
ar but quantitatively different
M lt 1 and M gt 1 Flows are qualitatively and
quantitatively different
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MEASUREMENT OF AIRSPEEDINCOMPRESSIBLE FLOW (M lt
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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MEASUREMENT OF AIRSPEEDSUBSONIC COMRESSIBLE
FLOW (0.3 lt M lt 1.0)

• If M gt 0.3, flow is compressible (density changes
  are important)
• Need to introduce energy equation and isentropic
  relations

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MEASUREMENT OF AIRSPEEDSUBSONIC COMRESSIBLE
FLOW (0.3 lt M lt 1.0)

• How do we use these results to measure airspeed?

• p0 and p1 give flight Mach number
• Instrument called Mach meter
• M1 V1/a1
• V1 is actual flight speed
• Actual flight speed using pressure difference
• What are T1 and a1?
• Again use sea-level conditions Ts, as, ps (a1
  (gRT)½ 340.3 m/s)
• V is called Calibrated Velocity, Vcal

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MEASUREMENT OF AIRSPEEDSUPERSONIC FLOW (M gt 1)
Rayleigh Pitot Tube Formula
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EXAMPLE SUBSONIC AND SUPERSONIC FLIGHT

• Flight at four different speeds, pitot measures
  p0 1.05, 1.2, 3 and 10 atm
• What is flight speed if flying in 1 atm static
  pressure and Tambient 288 K (a 340 m/s)?
• Determine which measurements are in subsonic or
  supersonic flow
• p0/p 1.893 is boundary between subsonic and
  sonic flows
• 1.05 atm ? p0/p 1.05 ? subsonic
• Use compressible flow form, M 0.265, V 90 m/s
  200 MPH
• Could use Bernoulli which will provide small
  error ( 1) and give V directly
• Compressible form requires knowledge of speed of
  sound (temperature)
• Apply Bernoulli safely? p0/p lt 1.065
• 1.2 atm ? p0/p 1.2 ? subsonic
• M 0.52, V 177 m/s 396 MPH
• Use of compressible subsonic form justified
  (Bernoulli 3 error)
• 3 atm ? p02/p1 3 ? supersonic
• M1 1.39, V 473 m/s 1057 MPH (Bernoulli
  22 error)
• 10 atm ? p02/p1 10 ? supersonic
• M1 2.73, V 928 m/s 2076 MPH (MCO ? LAX in 1
  hour 30 minutes)