WING LOADING (W/S), SPAN LOADING (W/b) AND ASPECT RATIO (b2/S) Span loading (W/b), wing loading (W/S) and AR (b2/S) are related Zero-lift drag, D0 is proportional to wing area Induced drag, Di, is proportional to square of span loading Take ratio of these drags, Di/D0 Re-write W2/(b2S) in terms of AR and substitute into drag ratio Di/D0 1: For specified W/S (set by take-off or landing requirements) and CD,0 (airfoil choice), increasing AR will decrease drag due to lift relative to zero-lift drag 2: AR predominately controls ratio of induced drag to zero lift drag, whereas span loading controls actual value of induced drag

EFFECT OF THICKNESS: NACA 4412 VS. NACA 4421 Both NACA 4412 and NACA 4421 have same shape of mean camber line Thin airfoil theory predict that linear lift slope and aL=0 should be the same for both Leading edge stall shows rapid drop of lift curve near maximum lift Trailing edge stall shows gradual bending-over of lift curve at maximum lift, “soft stall” High cl,max for airfoils with leading edge stall Flat plate stall exhibits poorest behavior, early stalling Thickness has major effect on cl,max

FINITE WING DOWNWASH a: Geometric Angle of Attack ai: Induced Angle of Attack aeff: Effective Angle of Attack Chord line Finite Wing Consequences: Tilted lift vector contributes a drag component, called induced drag (drag due to lift) → CL < cl and CD > cd Lift slope is reduced relative to infinite wing (a < a0)

IMPORTANT STATEMENTS Fundamental Equation of Thin Airfoil Theory “The camber line is a streamline of the flow” Fundamental Equation of Prandtl’s Lifting-Line Theory “The geometric angle of attack is equal to the sum of the effective angle of attack plus the induced angle of attack”

GENERAL LIFT DISTRIBUTION (2/4) Substitute expression for G(q) and dG/dy into fundamental equation of Prandtl’s lifting line theory Last term on the right (integral term) is a standard form and may be simplified as: Equation is evaluated at a given spanwise location (q0), just as fundamental equation of Prandtl’s lifting line theory is evaluated at a given spanwise location (y0) Only unknowns in equation are An’s Written at q0 equation is 1 algebraic equation with N unknowns Write equation at N spanwise locations to obtain a system of N independent algebraic equations with N unknowns