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Transcript
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)
2
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
3
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
4
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
5
EXAMPLE: FLAT PLATE
6
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
7
EXAMPLE: YF-102A vs. F-102A
8
EXAMPLE: YF-102A vs. F-102A
9
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)
10
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
11
SUPERCRITICAL AIRFOILS
12
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
13