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**MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS**

Summary of Incompressible Flow Over Airfoils Summary of Thin Airfoil Theory Example Airfoil Calculation Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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**KEY EQUATIONS FOR cl, aL=0, cm,c/4, and xcp**

Within these expression we need to evaluate A0, A1, A2, and dz/dx

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A0, A1, and A2 COEFFICIENTS

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**CENTER OF PRESSURE AND AERODYNAMIC CENTER**

Center of Pressure: It is that point on an airfoil (or body) about which the aerodynamic moment is zero Thin Airfoil Theory: Symmetric Airfoil: Cambered Airfoil: Aerodynamic Center: It is that point on an airfoil (or body) about which the aerodynamically generated moment is independent of angle of attack

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**ACTUAL LOCATION OF AERODYNAMIC CENTER**

x/c=0.25 NACA 23012 xA.C. < 0.25c x/c=0.25 NACA 64212 xA.C. > 0.25 c

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**EXAMPLE OF LEADING EDGE STALL**

NACA 4412 Airfoil (12% thickness) Linear increase in cl until stall At a just below 15º streamlines are highly curved (large lift) and still attached to upper surface of airfoil At a just above 15º massive flow-field separation occurs over top surface of airfoil → significant loss of lift Called Leading Edge Stall Characteristic of relatively thin airfoils with thickness between about 10 and 16 percent chord

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**EXAMPLE OF TRAILING EDGE STALL**

NACA 4421 (21% thickness) Progressive and gradual movement of separation from trailing edge toward leading edge as a is increased Called Trailing Edge Stall

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THIN AIRFOIL STALL Example: Flat Plate with 2% thickness (like a NACA 0002) Flow separates off leading edge even at low a (a ~ 3º) Initially small regions of separated flow called separation bubble As a increased reattachment point moves further downstream until total separation

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NACA 4412 VERSUS 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

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**OPTIMUM AIRFOIL THICKNESS**

Some thickness vital to achieving high maximum lift coefficient Amount of thickness will influence type of stalling behavior Expect an optimum Example: NACA 63-2XX, NACA looks about optimum NACA cl,max

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AIRFOIL THICKNESS

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**AIRFOIL THICKNESS: WWI AIRPLANES**

English Sopwith Camel Thin wing, lower maximum CL Bracing wires required – high drag German Fokker Dr-1 Higher maximum CL Internal wing structure Higher rates of climb Improved maneuverability

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**MODERN LOW-SPEED AIRFOILS**

NACA 2412 (1933) Leading edge radius = 0.02c NASA LS(1)-0417 (1970) Whitcomb [GA(w)-1] (Supercritical Airfoil) Leading edge radius = 0.08c Larger leading edge radius to flatted cp Bottom surface is cusped near trailing edge Discourages flow separation over top Higher maximum lift coefficient At cl~1 L/D > 50% than NACA 2412

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**MODERN AIRFOIL SHAPES Boeing 737 Root Mid-Span Tip**

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OTHER CONSIDERATIONS Note that all airfoils we have seen, even flat plate, will produce lift at some a Production of lift itself is not difficult L/D ratio Production of lift with minimum drag Measure of aerodynamic efficiency of wing or airplane Important impact on performance range, endurance Maximum lift coefficient, CL,max Effective airfoil shape produces high value of cl,max Stalling speed of aircraft (take-off, landing) Improved maneuverability (turn radius, turn rate)

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**HIGH LIFT DEVICES: SLATS AND FLAPS**

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**HIGH LIFT DEVICES: FLAPS**

Flaps shift lift curve Act as effective increase in camber of airfoil

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**AIRFOIL DATA: NACA 1408 WING SECTION**

Flap extended Flap retracted

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**HIGH LIFT DEVICES: SLATS**

Allows for a secondary flow between gap between slat and airfoil leading edge Secondary flow modifies pressure distribution on top surface delaying separation Slats increase stalling angle of attack, but do not shift the lift curve (same aL=0)

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RECALL BOEING 727 EXAMPLE cl ~ 4.5

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**EXAMPLE CALCULATION NACA 2412 Root Airfoil: NACA 2412**

GOAL: Find values of cl, aL=0, and cm,c/4 for a NACA 2412 Airfoil Maximum thickness 12 % of chord Maximum chamber of 2% of chord located 40% downstream of the leading edge of the chord line Check Out: NACA 2412 Root Airfoil: NACA 2412 Tip Airfoil: NACA 0012

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**EQUATIONS DESCRIBING MEAN CAMBER LINE: z = z(x)**

Equation describes the shape of the mean camber line forward of the maximum camber position (applies for 0 ≤ z/c ≤ 0.4) Equation describes the shape of the mean camber line aft of the maximum camber position (applies for 0.4 ≤ z/c ≤ 1)

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**EXPRESSIONS FOR MEAN CAMBER LINE SLOPE: dz/dx**

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**COORDINATE TRANSFORMATION: x → q, x0 → q0**

Equation describes the shape of the mean camber line slope forward of the maximum camber position Equation describes the shape of the mean camber line slope aft of the maximum camber position

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**EXAMINE LIMITS OF INTEGRATION**

Coefficients A0, A1, and A2 are evaluated across the entire airfoil Evaluated from the leading edge to the trailing edge Evaluated from leading edge (q=0) to the trailing edge (q=p) 2 equations the describe the fore and aft portions of the mean camber line Fore equation integrated from leading edge to location of maximum camber Aft equation integrated from location of maximum camber to trailing edge The location of maximum camber is (x/c)=0.4 What is the location of maximum camber in terms of q?

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**EXAMPLE: NACA 2412 CAMBERED AIRFOIL**

dcl/da = 2p Thin airfoil theory lift slope: dcl/da = 2p rad-1 = 0.11 deg-1 What is aL=0? From data aL=0 ~ -2º From theory aL=0 = -2.07º What is cm,c/4? From data cm,c/4 ~ From theory cm,c/4 =

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AIRFOIL WEB RESOURCES

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