MAE 3241 AERODYNAMICS AND FLIGHT MECHANICS Summary of

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 1

KEY EQUATIONS FOR cl, a. L=0, cm, c/4, and xcp • Within these expression we need to evaluate A 0, A 1, A 2, and dz/dx 2

A 0, A 1, and A 2 COEFFICIENTS 3

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 – Thin Airfoil Theory: • Symmetric Airfoil: • Cambered Airfoil: 4

ACTUAL LOCATION OF AERODYNAMIC CENTER x/c=0. 25 NACA 23012 x. A. C. < 0. 25 c x/c=0. 25 NACA 64212 x. A. C. > 0. 25 c 5

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 6

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 7

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 8

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 a. L=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 9

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 -2 XX, NACA 63 -212 looks about optimum NACA 63 -212 cl, max 10

AIRFOIL THICKNESS 11

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 12

MODERN LOW-SPEED AIRFOILS NACA 2412 (1933) Leading edge radius = 0. 02 c NASA LS(1)-0417 (1970) Whitcomb [GA(w)-1] (Supercritical Airfoil) Leading edge radius = 0. 08 c 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 13

MODERN AIRFOIL SHAPES Boeing 737 Root Mid-Span Tip http: //www. nasg. com/afdb/list-airfoil-e. phtml 14

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) 15

HIGH LIFT DEVICES: SLATS AND FLAPS 16

HIGH LIFT DEVICES: FLAPS • Flaps shift lift curve • Act as effective increase in camber of airfoil 17

AIRFOIL DATA: NACA 1408 WING SECTION Flap extended Flap retracted 18

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 a. L=0) 19

RECALL BOEING 727 EXAMPLE cl ~ 4. 5 20

EXAMPLE CALCULATION • GOAL: Find values of cl, a. L=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: http: //www. pagendarm. de/trapp/programming/java/profiles/ NACA 2412 Root Airfoil: NACA 2412 Tip Airfoil: NACA 0012 21

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) 22

EXPRESSIONS FOR MEAN CAMBER LINE SLOPE: dz/dx 23

COORDINATE TRANSFORMATION: x → q, x 0 → q 0 • 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 24

EXAMINE LIMITS OF INTEGRATION • Coefficients A 0, A 1, and A 2 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? 25

EXAMPLE: NACA 2412 CAMBERED AIRFOIL dcl/da = 2 p • Thin airfoil theory lift slope: dcl/da = 2 p rad-1 = 0. 11 deg-1 • What is a. L=0? – From data a. L=0 ~ -2º – From theory a. L=0 = -2. 07º • What is cm, c/4? – From data cm, c/4 ~ -0. 045 – From theory cm, c/4 = -0. 054 26

AIRFOIL WEB RESOURCES • • • http: //www. aerospaceweb. org/question/airfoils/q 0041. shtml http: //142. 26. 194. 131/aerodynamics 1/Basics/Page 4. html http: //www. aae. uiuc. edu/m-selig/ads. html http: //www. engr. utk. edu/~rbond/airfoil. html http: //www. nasg. com/afdb/index-e. phtml http: //www. pdas. com/avd. htm 27