Aerodynamics. Airfoils Kármán-Treftz mapping. The exponent k defines the internal angle of the trailing edge, τ, from. ( k)

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1 z = b Aerodynamics Kármán-Treftz mapping ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) The eponent defines the internal angle of the trailing edge, τ, from ( ) τ = π = = corresponds to the Jouowsi mapping z z + b b τ π ζ b = ζ + b

2 z = b Aerodynamics Kármán-Treftz mapping ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) = corresponds to a dividing streamline that leaves the trailing edge with tangential continuity. The trailing edge is not a stagnation point z z + b b ζ b = ζ + b τ = 0

3 z = b Aerodynamics Kármán-Treftz mapping ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) =1,95 corresponds to a dividing streamline that forms a dihedral at the trailing edge with an angle smaller than π. The trailing edge is a stagnation point z z + b b o τ = 9 ζ b = ζ + b

4 Kármán-Treftz mapping 1. Cylinder centre at the origin η z = b ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) y b ξ -b b =1,95, c = 3, 9b

5 η z b Aerodynamics Kármán-Treftz mapping. Cylinder centre on the imaginary ais = ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) y b ξ -b b =1,95, c = 3, 9b

6 η Aerodynamics Kármán-Treftz mapping 3. Cylinder centre on the real negative ais z = b ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) y b ξ -b- b = b ε ( 1+ ε ) ε = 1,95, c = 3, 9b +

7 η Aerodynamics Kármán-Treftz mapping 4. Cylinder centre on the nd quadrant z = b ( ζ + b) + ( ζ b) ( ζ + b) ( ζ b) y b ξ -b- b = b ε ( 1+ ε ) ε = 1,95, c = 3, 9b +

8 Kármán-Treftz mapping 1. Cylinder centre at the origin 1 -C p Etradorso, α=3 o Intradorso, α=3 o Etradorso, α=10 o Intradorso, α=10 o C p = p p r 1 ρ V /c

9 Kármán-Treftz mapping. Cylinder centre on the imaginary ais 3 -C p Etradorso, α=0 o Intradorso, α=0 o Etradorso, α=10 o Intradorso, α=10 o Etradorso, α=-10 o Intradorso, α=-10 o C p = p p r 1 ρ V /c

10 Kármán-Treftz mapping 3. Cylinder centre on the real negative ais Etradorso, α=0 o Intradorso, α=0 o Etradorso, α=10 o Intradorso, α=10 o -C p C p = p p r 1 ρ V /c

11 Kármán-Treftz mapping 4. Cylinder centre on the nd quadrant 4.5 -C p Etradorso, α=0 o Intradorso, α=0 o Etradorso, α=10 o Intradorso, α=10 o Etradorso, α=-10 o Intradorso, α=-10 o C p = p p r 1 ρ V /c

12 Generalization of the conformal mapping The Jouowsi transform is one of the mappings defined by z = ζ + n= 1 a n n ζ For the Jouowsi case: a1 = b an = 0 para n

13 Generalization of the conformal mapping z = ζ + n= 1 a n n ζ In general, the coefficients a n are comple Theoretically, the generalized mapping is able to obtain any airfoil from a cicular cylinder

14 Generalization of the conformal mapping z = ζ + n= 1 At small angles of attac, the lift coefficient, C l, of an airfoil in steady, irrotacional and incompressible flow is given by C For a Jouowsi airfoil a n n ζ d = π + at + c l 1 ( α β ) a t 0,77

15 M For an airfoil Q=0 Aerodynamics Pitching moment relative to the airfoil centre QΓ = ρ π + R Μ e 0 M 0 [ ] iα i πρu [ ] iα i πρ U = R e Μ Μ is the term proportional to z- of the comple velocity at large distances from the airfoil Assuming that the a 1 coefficient is given by a 1 iλ = b e Μ = a r r iα iα 1 V e a V e

16 Pitching moment relative to the airfoil centre For small values of α M c r 4πρ b V + ( α λ) Assuming a chord c 4b C M c = c M r ρ V 1 c π ( α + λ)

17 NACA airfoils The development of the NACA airfoils started in 1933 at the National Advisory Committee for Aeronautics (NACA), designated National Aeronautics and Space Administration (NASA) in nowadays NACA airfoils are obtained from the addition of a thicness distribution to a camber line. Several series of airfoils have been developed along the years

18 NACA airfoils 4 digits series, NACA ABCD A Maimum relative camber in percentage, f/c B Coordinate of maimum camber, m, given by 10 m /c CD Maimum relative thicness in percentage, d/c

19 NACA airfoils 4 digits series, NACA ABCD y - Thicness distribution d = 0, ( ) 3 4 0,969 0,16 0, , Camber lines parabola arcs matching at the point of maimum camber, located at m

20 NACA airfoils 4 digits series, NACA ABCD y - Thicness distribution d y = 0, m f = m 1 m ( ) 3 4 0,969 0,16 0, , Camber lines f ( ) m ( ) ( 1 )( 1+ ) m m m > m A f = 100 m B = 10

21 NACA airfoils 4 digits series - NACA 001 Symmetric airfoil with 1% of maimum relative thicness

22 NACA airfoils 4 digits series - NACA 441 Airfoil with maimum relative camber of 4% located at m =0.4c (40% from the leading edge) and maimum relative thicness of 1%

23 5 digits series, NACA ABCDE A Approimate value of BC Coordinate of maimum camber, m, given by 00 m /c DE Maimum relative thicness in percentage, d/c ( C l ) proj NACA airfoils 10 3 is the project lift coefficient ( C l ) proj

24 NACA airfoils 5 digits series, NACA ABCDE y - Thicness distribution identical to 4 digits series d = 0, ( ) 3 4 0,969 0,16 0, , Camber line polynomials with decreasing curvature from starting at the leading edge. Straight line to the right of the maimum camber point, located at m

25 NACA airfoils 5 digits series, NACA ABCDE - Thicness distribution identical to 4 digits series d y = 0, - Camber line 1 6 = 1 6 y ( ) 3 4 0,969 0,16 0, , m 1 1 ( 3 m) ( 3 3m + m ) m 3 ( 1 ) m > m m

26 Aerodynamics 5 digits series, NACA ABCDE Designação m NACA airfoils - Camber line Values of 1, m e m for C l = ( ) 0, 3 proj m ,05 0, ,4 0 0,10 0,160 51, ,15 0,05 15, , , ,5 0,3910 3,30

27 NACA airfoils 5 digits series - NACA 3015 ( ) 3 Airfoil with C l = 0,, maimum camber located proj at m =0.15c (15% from the leading edge) and 15% of maimum relative thicness

28 NACA airfoils 6 digits series, NACA 6A,B-CDE, a=a o A Value of p 10c. p is the horizontal coordinate of the suction pea of the corresponding symmetric foil at zero degrees angle of attac. p is related to the thicness distribution of the airfoil

29 NACA airfoils 6 digits series, NACA 6A,B-CDE, a=a o B Range of C l values (multiplied by 10) above and below the value of C l for project conditions, which correspond to favourable or close to zero pressure gradients on the upper and lower sides of the airfoil. 10 ( C l ) proj ( C l ) proj C. is the lift coefficient for project conditions

30 NACA airfoils 6 digits series, NACA 6A,B-CDE, a=a o DE Maimum relative thicness in percentage a o Maimum horizontal coordinate of the region that presents an approimately constant loading (pressure difference between upper and lower sides of the airfoil). For >a o the loading decreases linearly. a o is related to the camber line NACA (a=1)

31 y M V r BA α Pitching moment relative to the leading edge L α M c - Moment propagation M C M BA = M = C c M + L cos + C l ( α ) cos c M 1 c c + L ( α ) CM + Cl BA c c 1

32 y M V r BA α Pitching moment relative to the leading edge L α M c - For an airfoil at small values of α π C l π C M + c so π π CM ( β λ) + ( α + β ) BA π Cl CM γ + com γ = β λ BA 4 ( α + β ) ( α λ)

33 Aerodynamic centre L α y M - V r α M c Aerodynamic centre is the location that ehibits a pitching moment independent of the angle of attac, α Pitching moment relative to a point CM CM C c l c

34 Aerodynamic centre L α y M - V r α M c The location of the aerodynamic centre is given by dcm dc M = 0 = 0 so dα dcl dcm dc dc ca ca M M 1 c c c = 0 = = dc c c dc dα dc l l l dα

35 Aerodynamic centre L α y M - V r α M c For an airfoil at small values of α dcm c π dcl π dα dα so ca c π 1 = π 1 4

36 Aerodynamic centre L α y M - V r α M c The aerodynamic centre is approimately located 5% of the chord downstream of the leading edge The pitching moment relative to the aerodynamic centre is given by dcm dc π c CM = ca dα dα M c ( λ β ) = γ γ

37 Pressure centre L α y M - V r α M c Pressure centre is the point along the line that contains the chord that ehibits a zero pitching moment CM CM Cl = 0 c c

38 V r α Pressure centre L α y M c M cp C CM CM Cl = 0 = c c c C For an airfoil at small values of α π C l π C M + c so cp 1 1 γ 1 πγ = + = + = c 4 4 α + β 4 C ( α + β ) ( α λ) l M l c C - M C l ca

Introduction to Aeronautics

Introduction to Aeronautics ARO 101 Sections 03 & 04 Sep 30, 2015 thru Dec 9, 2015 Instructor: Raymond A. Hudson Week #8 Lecture Material 1 Topics For Week #8 Airfoil Geometry & Nomenclature Identify the