1. 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

2. 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
Nature of PDE:
Subsonic: (1 - M∞2) > 0 (elliptic)
Supersonic: (1 - M∞2) < 0 (hyperbolic)

3. 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

4. DERIVATION OF PRESSURE COEFFICIENT, CP
Solutions to hyperbolic wave equation
Velocity perturbations
Eliminate f’
Linearized flow tangency condition at surface
Linearized definition of pressure coefficient
Combined result
Positive q: measured above horizontal
Negative q: measured below horizontal

5. 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

6. EXAMPLE: FLAT PLATE

7. 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

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

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

10. 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)

11. 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

12. SUPERCRITICAL AIRFOILS

13. 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