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MAE 3241: AERODYNAMICS AND
FLIGHT MECHANICS
Compressible Flow Over Airfoils:
Linearized Supersonic Flow
Mechanical and Aerospace Engineering Department
Florida Institute of Technology
D. R. Kirk
SMALL PERTURBATION VELOCITY POTENTIAL EQUATION
• Equation is a linear PDE and easy to solve
1  M 
uˆ vˆ
 0
x y
1  M 
2 ˆ
ˆ
 
 2 0
2
x
y
2

2

2
• 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
• Nature of PDE:
– Subsonic: (1 - M∞2) > 0 (elliptic)
– Supersonic: (1 - M∞2) < 0 (hyperbolic)
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SUPERSONIC APPLICATION
1  M 
2

2 ˆ
ˆ
 
 2 0
2
x
y
2
2 ˆ
ˆ

2  
l
 2 0
2
x
y
2
• Linearized small perturbation equation
• Re-write for supersonic flow
l  M 2  1
ˆ  f  x  ly 
• 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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DERIVATION OF PRESSURE COEFFICIENT, CP
• Solutions to hyperbolic wave equation
ˆ  f  x  ly 
ˆ
uˆ 
 f ;
x
vˆ
uˆ  
ˆ
vˆ 
 lf  • Velocity perturbations
y
l
ˆ
vˆ 
 V tan   V
y
V
ˆu   
• Eliminate f’
• Linearized flow tangency condition at surface
l
2uˆ
CP  
V
CP 
2
M 2  1
• Linearized definition of pressure coefficient
• Combined result
– Positive : measured above horizontal
– Negative : measured below horizontal
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KEY RESULTS: SUPERSONIC FLOWS
CP 
cl 
cd 
2
• Linearized supersonic pressure coefficient
M 2  1
4
M 2  1
4
2
M 2  1
• 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
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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 crosssectional 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
– Concavity in rear part of pressure side
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SUPERCRITICAL AIRFOILS
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SUPERCRITICAL AIRFOILS
1. For given thickness, supercritical airfoil allows for higher cruise velocity
2. 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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