Lifting Surfaces. Lifting Surfaces

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Edward Lawson
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1 Lifting Sufaces A lifting suface geneates a foce pependicula to the undistued flow, lift foce, much lage than the foce in the diection of the undistued flow, dag foce. L D Aeodynamic foce Dag Lifting Sufaces The typical example of a lifting suface is the wing of an aiplane. The lades of a popelle o an axial tuomachine o the aileons of acing cas ae also lifting sufaces

2 Lifting Sufaces The classic aeodynamic theoy of lifting sufaces divides the polem in two pats: - Two-dimensional of the section (aifoil) - Tip effects Finite wings Lifting Sufaces Nomenclatue - Span - S Pojected aea - Λ Aspect Ratio Λ S - Coss-section (Ai)foil

3 Aifoils Lifting Sufaces Nomenclatue - L Leading edge - T Tailing edge - c Chod Aifoils Lifting Sufaces Nomenclatue - Came line contains the centes of all cicles inscied in the aifoil - Maximum thickness, d, is the maximum diamete of the inscied cicles - Relative thickness is the atio etween the maximum thickness and the chod, d/c

4 Aifoils Lifting Sufaces Nomenclatue - Maximum came is the maximum distance etween the came line and the staight line that connects the edges of the came line (chod) V - Angle of attack, α, is the angle etween the diection of the undistued flow,, and the chod V Aifoils Lifting Sufaces Nomenclatue

5 Lifting Sufaces Nomenclatue Lift, L. Foce pependicula to the diection of the undistued flow U C C L l L ρ V L ρ V S c (3 D) ( D) V Lifting Sufaces Nomenclatue Dag, D. Foce in the diection of the undistued flow U C C D d D ρ V D ρ V S c (3 D) ( D) V

6 Aifoils Joukowski tansfom is a confomal mapping that tansfoms a cicula cylinde into an aifoil accoding to the following expession(s) z z + ( ) ( + ) + z z + z + Aifoils z + The aifoil is in plane z (tansfomed plane) The cicula cylinde is in plane (stating plane) The cylinde is a continuous line. The only way to geneate the tailing edge (cone) is to guaantee that the point that is tansfomed to the tailing edge () is a mapping singulaity

7 Mapping deivative Aifoils z + Mapping singulaities ae located at dz d dz 0 ± d Aifoils z + Mapping singulaities ae located at dz 0 ± d The tailing edge is otained fom the point in the plane. is the distance etween the intesection of the cylinde with the positive eal axis and the oigin of the coodinate system

8 Aifoils Kutta condition Velocity in aifoil plane (z) is otained fom dw dz dw d dz d Velocity in the cylinde plane Mapping deivative The point in the stating plane must e a stagnation point to avoid an infinite velocity at the tailing edge Aifoils Kutta condition The point in the stating plane must e a stagnation point to avoid an infinite velocity at the tailing edge dw dz odo de fuga dw d dz d 0 0 Kutta condition states that the velocity at the tailing edge must e finite. This condition defines the ciculation of the flow aound the aifoil

9 Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ - Complex potential fo a coodinate system with the eal axis aligned with the undistued flow and the cente of the cylinde at the oigin W a Γ V + i ln π ( ) Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ - Complex velocity fo a coodinate system with the eal axis aligned with the undistued flow and the cente of the cylinde at the oigin dw a Γ V V i d π

10 Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ - Stagnation points Γ z i 4π V ± a Γ 4πa V - It only makes sense to conside Γ < 4πa V Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ - The agument of the stagnation points, θ, is given y Γ sen( θ ) 4πaV Γ 4πaV sen ( θ ) θ θ Γ < 0

11 Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ - The agument of the stagnation points, θ, is given y Γ > 0 Γ sen( θ ) 4πaV Γ 4πaV sen ( θ ) θ θ Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ. Cylinde cente at the oigin

12 Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ. Cylinde cente on the imaginay axis Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ 3. Cylinde cente on the negative eal axis

13 Aifoils Stating flow in plane is the flow aound a cicula cylinde of adius a with ciculation, Γ 4. Cylinde cente on the nd o 3 d quadants Aifoils. Cylinde cente at the oigin o 0 + i 0, a - Mapping of the geomety e iθ iθ z e + e z cos iθ ( θ ) The cicumfeence is tansfomed into a flat plate with a length of 4

14 Aifoils. Cylinde cente at the oigin η z + y ξ - x Aifoils. Cilinde cente at the oigin - Kutta condition must e a stagnation point in the cylinde plane () η Γ 4πa V sen( α ) Γ α ξ

15 Aifoils. Cylinde cente at the oigin α 0 º, Γ 0 Aifoils. Cylinde cente at the oigin α 0º, Γ 4πa V sen ( ) α

16 . Cylinde cente at the oigin - Lift coefficient, Cl Aifoils L ρ V Γ Cl ρ V ρ c V c Γ 4πaV sen, c 4, C ( α ) π sen ( ) α - Fo small angles of attack πα l C l Γ Γ V c a sen( α ) α Aifoils. Cylinde cente at the oigin - Velocity at the tailing edge dw d ( V ) odode fuga dz d - Solving the indetemination U V ( V ) odode fuga V cos( α ) V cos ( α )

17 . Cylinde cente at the oigin - Pessue distiution on the plate suface -C p Aifoils x/c Extadoso, α3 o Intadoso, α3 o Extadoso, α0 o Intadoso, α0 o C p p p ρ V Aifoils. Cylinde cente on the imaginay axis ( β ), cos( β ) 0 + i asen a o η β a ξ

18 Aifoils. Cylinde cente on the imaginay axis η z + y ξ - f β f f tan ( β ) c c x The cicumfeence is tansfomed into a cameed plate (ac of a cicle) with chod 4 Aifoils. Cylinde cente on the imaginay axis - Kutta condition must e a stagnation point in the cylinde plane () Γ 4π av sen( α + β ) η Γ β β α ξ

19 Aifoils. Cylinde cente on the imaginay axis f α 0º, β 0º 0, 088 c Aifoils. Cylinde cente on the imaginay axis f α 0º, β 0º 0, 088 c

20 Aifoils. Cylinde cente on the imaginay axis f α 0º, β 0º 0, 088 c Aifoils. Cylinde cente on the imaginay axis - Lift coefficient, C l L ρ V Γ Γ Γ Cl ρ V ρ c V c V c Γ 4πaV sen α + β, c 4, a cos β C l ( ) ( ) sen( α + β ) π cos( β )

21 . Cylinde cente on the imaginay axis - Lift coefficient, Aifoils C l - Fo small angles of attack (α) and small values of β sen α + β α + β, cos β ( ) ( ) C l ( α β ) π + - The cuvatue poduces an hoizontal tanslation of β in the line C l f ( α ). Fo the same angle of attack a cameed plate geneates a lage lift than a flat plate. Cylinde cente on the imaginay axis - Lift coefficient, Cl Aifoils C l ( α β ) π + ( ) - A cameed plate poduces lift C l 0 fo an angle of attack of zeo degees, without exhiiting stagnation points and suction peak - The angle α β is the zeo lift angle

22 Aifoils. Cylinde cente on the imaginay axis - Pessue distiution on the plate suface -C p x/c Extadoso, α0 o Intadoso, α0 o Extadoso, α0 o Intadoso, α0 o Extadoso, α-0 o Intadoso, α-0 o C p p p ρ V Aifoils 3. Cylinde cente on the negative eal axis ( + ) a o ε + i 0, ε η ε a ξ

23 Aifoils 3. Cylinde cente on the negative eal axis η z + y d c 3 3 ε 4 ξ -- d x ε 4 + ε The cicumfeence is tansfomed into a symmetic aifoil with chod ε ε Aifoils 3. Cylinde cente on the negative eal axis - Kutta condition must e a stagnation point in the cylinde plane () η Γ 4πa V sen( α ) Γ α ξ

24 Aifoils 3. Cylinde cente on the negative eal axis d α 0º, ε 0,5 0, 95 c Aifoils 3. Cylinde cente on the negative eal axis d α 0º, ε 0,5 0, 95 c

25 Aifoils 3. Cylinde cente on the negative eal axis C l - Lift coefficient, C l L ρ V Γ ρ V ρ c V c Γ 4πaV Γ Γ V c ε sen( α ), c 4 +, ε + ε Cl π + sen( α ) + ε a + ε 3. Cylinde cente on the negative eal axis - Lift coefficient, Cl Aifoils d C l π + 0,77 sen c - Fo small angles of attack ( ) α sen( α ) α d C l π + 0, 77 α c - The thickness inceases the slope of C l f ( ) α

26 Aifoils 3. Cylinde cente on the negative eal axis - Pessue distiution on the aifoil suface -C p x/c Extadoso, α0 o Intadoso, α0 o Extadoso, α0 o Intadoso, α0 o C p p p ρ V ε + i asen Aifoils 4. Cylinde cente on the nd quadant ( β ), ( + ε ) a cos( β ) o η β ε a ξ

27 Aifoils 4. Cylinde cente on the nd quadant η z + y d c 3 3 ε 4 ξ -- x ε f f 4 tan ( β ) + ε c c The cicumfeence is tansfomed into a asymmetic aifoil with chod ε ε Aifoils 4. Cylinde cente on the nd quadant - Kutta condition must e a stagnation point in the cylinde plane () Γ 4π av sen( α + β ) η Γ α β β ξ

28 Aifoils 4. Cylinde cente on the nd quadant f d α 0º, β 0º 0,088, ε 0,5 0, 95 c c Aifoils 4. Cylinde cente on the nd quadant f d α 0º, β 0º 0,088, ε 0,5 0, 95 c c

29 Aifoils 4. Cylinde cente on the nd quadant 95 0, 0,5, 0,088 0º 0º, c d c f ε β α 4. Cylinde cente on the nd quadant - Lift coefficient, l C ρ Γ Γ V L Aifoils ( ) ( ) ( ) ( ) β β α ε ε π ε β ε ε β α π ρ ρ ρ cos sen cos, 4, sen Γ Γ Γ l l C a c av c V c V V c V L C

30 4. Cylinde cente on the nd quadant - Lift coefficient, Aifoils C l - Fo small angles of attack (α) and small values of β sen α + β α + β, cos β ( ) ( ) d C l π + 0, 77 + c ( α β ) - The aifoil includes came and thickness effects: - Hoizontal tanslation of β in the line C l f ( α ) - Incease of the slope of C l f α ( ) Aifoils 4. Cylinde cente on the nd quadant - Pessue distiution on the aifoil suface -C p x/c Extadoso, α0 o Intadoso, α0 o Extadoso, α0 o Intadoso, α0 o Extadoso, α-0 o Intadoso, α-0 o C p p p ρ V

31 Aifoils.5 Vaiation of C l with α.5 C l α (Gaus) Aifoils Pitching moment elative to the aifoil cente yη C C is the aifoil cente xξ iα [ i πρu ] QΓ M 0 ρ + R Μe π Fo a Joukowski aifoil Q0 M 0 R iα [ i πρu ] Μe

32 Aifoils Pitching moment elative to the aifoil cente Μ is the tem popotional to z - of the complex velocity at lage distances fom the aifoil Μ V e a V e iα iα so M c πρ V sen α ( ) Aifoils Pitching moment elative to the aifoil cente Fo small values of α ( ) α M c sen α 4 πρ V α Assuming that a Joukowski aifoil as a chod appoximately equal to c 4, the pitching moment aound the aifoil cente is given y C M c M c π α ρ V c

Lifting Surfaces. Lifting Surfaces

Lifting Surfaces. Lifting Surfaces Lifting Sufaces A lifting suface geneates a foce pependicula to the undistued flow, lift foce, much lage than the foce in the diection of the undistued flow, dag foce. L D Aeodynamic foce Dag Lifting Sufaces