MAE 3241 AERODYNAMICS AND FLIGHT MECHANICS Compressible Flow

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) 2

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 3

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 4

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 – 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 2 nd derivative of volume distribution of vehicle 7

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

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