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 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)

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

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

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

EXAMPLE: FLAT PLATE

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

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

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

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)

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

SUPERCRITICAL AIRFOILS

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