Aerodynamics. High-Lift Devices

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1 High-Lift Devices Devices to increase the lift coefficient by geometry changes (camber and/or chord) and/or boundary-layer control (avoid flow separation - Flaps, trailing edge devices - Slats, leading edge devices

2 High-Lift Devices Flaps Flat plate deflected at the trailing edge for zero angle of attack Loading distribution for an ideal fluid flowing around a flat plate with a deflected trailing edge C l > 0 due to the camber introduced by the trailing edge deflection. Adverse pressure gradients lead to flow separation

3 High-Lift Devices Flaps Flat plate deflected at the trailing edge for zero angle of attack or Separation at the flap corner Boundary-layer separation reduces the stall angle of attack. Boundary-layer control may avoid the reduction of the stall angle of attack

4 Plain flap High-Lift Devices Flaps a) Clean configuration b) Deflected flap

5 Plain flap High-Lift Devices Flaps Boundary-layer separation reduces the stall angle of attack

6 Flap split High-Lift Devices Flaps Simple deflection of a plate. Increase of C l is similar to plain flap, but there is a significant increase of C d compared to the plain flap

7 Slotted flap Aerodynamics High-Lift Devices Flaps Boundary-layer control at the trailing edge. Separation delayed by blowing at the region of adverse pressure gradient on the upper side of the flap

8 Fowler flap High-Lift Devices Flaps Boundary-layer control similar to the slotted flap. Increase of C l due to growth of the chord length C l L L = r = r 1 2 ρ V c 1 2 ρ V c f cf c = C 2 2 lf cf c

9 Lift coefficient Aerodynamics High-Lift Devices Flaps Fowler flap Slotted flap Plain flap Basic airfoil (no flaps)

10 High-Lift Devices Flaps Lift and drag coefficients

11 High-Lift Devices Leading edge slats Boundary-layer control at the suction peak close to the airfoil leading edge a) Basic section: separation bubble at the leading edge b) Deflected leading edge

12 High-Lift Devices Leading edge slats Boundary-layer control at the suction peak close to the airfoil leading edge Slot at the leading edge

13 High-Lift Devices Examples Fowler flap with double slot

14 High-Lift Devices Examples Deflection of high-lift devices in different flight configurations

15 High-Lift Devices Examples

16 Aerodynamic Appendages Spoiler

17 Aerodynamic Appendages

18 Aerodynamics For a steady, irrotational and incompressible flow around an airfoil it is possible to define a velocity potential function, φ, that satisfies the Laplace equation r r φ = 0 and the following boundary conditions r r φ φ n = = vw em S n r r φ = V para r

19 r r φ φ n = = vw on S n r r φ = V for r - V r is the velocity of the incoming flow assumed to be uniform v w - is the normal velocity component on the body surface that is equal to 0 for an impermeable surface

20 The velocity potencial function, φ, is split (linear problem) in to the velocity potential of the uniform incoming flow, φ, and the pertubation potential, Φ, that represents the effect of the airfoil φ = φ + Φ φ is known and Φ is defined by a superficial distribution of sources/sinks lines of intensity σ(q)

21 If the mathematical model contains only a uniform flow and sources/sinks distributions it has no circulation, Γ=0. Therefore, it can not satisfy the Kutta condition

22 The circulation required to satisfy the Kutta condition is introduced by the velocity potential of a purely circulatory flow, φ Γ

23 The circulatory flow (function φ Γ ), can be defined with superficial and/or interior line vortices distributions

24 The circulation introduced depends on a constant, γ o, that must be determined to satisfy the Kutta condition

25 The velocity induced by the vortex distribution at a point P is designated by V r γ, where V r o Γ Γ is the velocity induced by the vortex distribution with γ o =1

26 Aerodynamics The normal velocity component at any point P of the airfoil surface is obtained from φ, Φ e φ Γ and it satisfies the following equation σ 2 ( P) 1 [ ln( r( P, q) )] σ ( q) ds + ( V ) + γ ov np = vwp + Γ S 2π np - r(p,q) stands for the distance between point P and a point q on the airfoil surface - n r P is the (external) normal vector to the airfoil surface at point P r r r

27 ( P) σ 2 1 S 2π n r r V np r r γ V n v o w P Γ P P [ ln( r( P, q) )] σ ( q) ds Self-induction Normal velocity component induced by the superficial distribution of source/sinks lines Normal velocity component induced by the uniform incoming flow Normal velocity component induced by the circulatory flow Normal velocity component at point P

28 σ 2 ( P) 1 [ ln( r( P, q) )] σ ( q) ds + ( V ) + γ ov np = vwp + Γ S 2π np r r r The application of the boundary conditions on the airfoil surface leads to a Fredholm integral equation of the second kind that defines the intensity σ(q) of the line sources/sinks distribution

29 σ 2 ( P) 1 [ ln( r( P, q) )] σ ( q) ds + ( V ) + γ ov np = vwp + Γ S 2π np r r r The constant γ o is determined by the application of the Kutta condition. At the velocity at the trailing edge must be finite, i.e. the dividing streamline must be aligned with the bisector of the trailing edge angle

30 σ 2 ( P) Aerodynamics 1 [ ln( r( P, q) )] σ ( q) ds + ( V ) + γ ov np = vwp + Γ S 2π np The panel method transforms the Fredholm integral equation of the second kind into a system of algebraic equations that as a straighforward numerical solution. There are two types of approximations required: 1. Geometric representation of the airfoil shape 2. Piecewise (panels) definition of σ(q) r r r

31 Geometric discretization - The airfoil is described by n flat elements (panels) defined by boundary points. When n the approximate geometry converges to the exact shape of the airfoil Boundary points Control points

32 Approximation of σ(q) - For each flat element (panel), the intensity of the source/sinks lines remains constant. When n the approximate description of σ(q) tends to a continuous (exact) distribution Boundary points Control points

33 Boundary conditions - The boundary conditions (specification of the normal velocity component) are satisfied at the midpoint of each panel (control point). When n the boundary conditions are satisfied for the complete airfoil Boundary points Control points

34 Aerodynamics The discretization process leads to the following system of algebraic equations (n (n+1)): n A + r r ( V V ) r ijσ j + γ o ni = vw with 1 i n Γ j= 1 - Unknowns: n values of σ and γ o, total of n+1 - Equations: n control points to impose the normal component of the velocity The n+1 equation is the Kutta condition i i

35 Aerodynamics The discretization process leads to the following system of algebraic equations (n (n+1)): n A + r r ( V V ) r ijσ j + γ o ni = vw with 1 i n Γ j= 1 i - A ij is the influence coefficients matrix that define the normal velocity component induced at point i by a line source distribution of intensity one in panel j. - Analogously, the matrix B ij is defined for the tangential velocity component i

36 The influence coefficients matrices, A ij e B ij, are exclusively dependent on the airfoil geometry and can be easily determined in the local reference frames of the elements (panels) - Calculation of A ij and B ij 1. Transform the coordinates x,y to a reference frame with the horizontal axis aligned with panel j and the vertical axis aligned with the external normal of the (approximate) airfoil

37 - Calculation of A ij and B ij 2. Determine the induced velocity in the local reference frame V V x y = = 1 2π 1 2π L L E E 0 2 xe ζ dζ r ye dζ r 0 2 L is the length of E panel j

38 E L is the length of panel j ( ) = + + = e E e e e y e E e e e x y L x y x V y L x y x V arctg arctg 2 1 ln π π - Calculation of A ij and B ij 2. Determine the induced velocity in the local reference frame

39 - Calculation of A ij and B ij 3. Project the velocity vector obtained in the local reference frame of panel j to a reference frame with the horizontal axis aligned with panel i and with the vertical axis aligned with the external normal of the (approximate) airfoil at panel i

40 Aerodynamics There are several alternatives to impose the Kutta condition numerically: 1. Equal velocity (pressure) at the two control points of the panels that define the trailing edge 2. Equal velocity (pressure) at the trailing edge for extrapolations of the velocity based on the nearby control points 3. Define an extra control point downstream of the trailing edge and impose that the velocity normal to the bisector of the trailing edge is equal to zero

41 Aerodynamics Any of the previous alternatives leads to the extra algebraic equation that determines γ o and consequently Γ 1. Simple implementation, numerically robust and sufficiently accurate for reasonable panel sizes at the trailing edge (cosine distribution of boundary points along the chord) 2. The extra equation only remains linear for linear extrapolations. In that case, differences to the previous option are not significant

42 Any of the previous alternatives leads to the extra algebraic equation that determines γ o and consequently Γ 3. The distance of the extra control point to the trailing edge is ambiguous. Too small distances lead to numerical difficulties and too large distances to a loss of accuracy. Ideal distance depends on the local geometry of the airfoil (camber and trailing edge angle)

43 Theoretically, any superficial or interior vortex distribution may be used to satisfy the Kutta condition Numerical experiments show that superficial or internal vortex distributions lead to different results. Although the value of Γ may be correct, there are alternatives that lead to an incorrect pressure distribution at the trailing edge

44 Linear along the mean line Constant on the surface Parabolic on the surface Analytic solution Numerical solutions

s m Aerodynamics - is the distance to the trailing edge measured along the mean line panels - m is the number of mean line panels ( m = n 2) - is the length of the mean line panels l m

45 s m Aerodynamics - is the distance to the trailing edge measured along the mean line panels - m is the number of mean line panels ( m = n 2) - is the length of the mean line panels l m Vortex distribution along de mean line of the airfoil with an intensity given by γ = γ s o 0,4 m (constant γ s o 0,4 m i in each panel) Boundary points Control points Mean line boundary points

46 Vortex distribution along de mean line of the airfoil with an intensity given by γ = γ s o 0,4 m (constant γ s o 0,4 m i in each - The total circulation around the airfoil is given by Γ = panel) L m m γds = γ o 0 i= 1 s 0,4 m i l Boundary points Control points Mean line boundary points m i

47 σ j and γ o are determined from the solution of the system of n+1 algebraic equations The tangential velocity components at the control points are given by n r r v B ( V V ) r t = ijσ j + + γ o Γ ti with 1 i n i j= 1 - t r i is the unit vector tangent to panel i i

48 The pressure coefficient is defined by p p C p = r 2 2 ρ V 1 In irrotational, incompressible flow (Bernoulli equation) r 2 1 V C p = r V

49 At the control points n Aerodynamics r v = 0 V = v i t i C p = 1 vt r i V 2

50 Aerodynamics Determination of the forces - There are two alternatives to determine the lift coefficient: Kutta-Joukowski equation (circulation) or integration of the surface pressure distribution - For an ideal fluid, the drag coefficient is zero. However, the numerical solution will only respect this result when the number of panels tends to infinity. The drag coefficient obtained from the integration of the surface pressure distribution is a useful measure of the discretization error/uncertainty

51 - Lift coefficient Aerodynamics Determination of the forces C l = 2Γ r V c ou C l = C y cos ( α ) C sen( α ) x

52 - Drag coefficient Aerodynamics Determination of the forces C = C cos + d x ( α ) C sen( α ) y

53 = = = = + + n i pf pf p p y n i pf pf p p x c x x C c x d C C c y y C c y d C C i i i i i i For flat panels and C p determined at the control points Determination of the forces

54 Aerodynamics Including viscous effects - Viscosity originates boundary-layers on the upper and lower surfaces of the airfoil (and a wake). The streamlines of the external are displaced δ * as a consequence of this viscous region of the flow - Boundary-layer calculations to determine δ* require the knowledge of the pressure gradient, which depends on the displacement thickness,δ *. Therefore, the solution procedure must be iterative (not possible if there is flow separation)

55 Including viscous effects - Two possible alternatives: 1. Define an aerodynamic shape of the airfoil adding the displacement thickness to the airfoil geometry (geometry change) 2. Keep the airfoil geometry fixed and introduce the δ * effect with a transpiration velocity, v w, at the airfoil surface

56 Aerodynamics Including viscous effects - Two possible alternatives: 1. Change of geometry implies the re-calculation of influence coefficients matrics everry iteration 2. Fixed geometry implies a fixed matrix of coefficients for the linear system that can be factorized. Introduction of the δ * effect corresponds to a simple change of the right-hand side

57 Including viscous effects - Determination of the transpiration velocity = y v v dy 0 y - From the continuity equation v = y u 0 x dy

58 Including viscous effects - Determination of the transpiration velocity = y u v dy 0 x - Adding and subtracting the external velocity to the boundary-layer U e = v 0 y x U e U e 1 u U e dy

59 ( ) * δ δ e e e e e y e e e U dx d x U v dy U u U dx d x U v dy U u U U x v + = + = = Including viscous effects - Determination of the transpiration velocity

60 Including viscous effects - Determination of the transpiration velocity U e d v = + U e δ x dx ( *) - The first term is of inviscid nature and the second one include the boundary-layer effect, so d v w = U e δ dx ( *)

61 ASA2D program ASA2D is a computer code for the numerical calculation of the aerodynamic characteristics of airfoils in steady, incompressible flows at small angles of attack (no flow separation) Weak viscous-inviscid interaction method that can not handle flow separation Iterative solution procedure with fixed geometry and transpiration velocity to include viscous effects in the ideal fluid solution

62 ASA2D program Adopted methods - Ideal fluid flow (potential flow) - Boundary-layer (viscous flow) Thwaites s method (laminar regime) Head s method (turbulent regime) Instantaneous transition at a point

63 Adopted methods Aerodynamics ASA2D program - Boundary-layer (viscous flow) Correlations H Re and R for the determination x e R θ e x of the transition point Wake influence neglected - Boundary-layer separation for laminar flow is assumed to be a transition criterion to turbulent flow

64 ASA2D program Determination of forces and moments - Lift coefficient, C l Kutta-Joukowski equation (circulation) Integration of the pressure distribution

65 ASA2D program Determination of forces and moments - Drag coefficient, C d Integration of pressure and wall shear-stress on the airfoil surface Squire & Young formula C d 2 = θbf e c r V r V bf H bf e θ bf i r V r V bf H bfi 2 + 5

66 ASA2D program Determination of forces and moments - Pitching moment relative to the airfoil centre coefficient Integration of pressure distribution

1. Fluid Dynamics Around Airfoils

1. Fluid Dynamics Around Airfoils Two-dimensional flow around a streamlined shape Foces on an airfoil Distribution of pressue coefficient over an airfoil The variation of the lift coefficient with the