Incompressible Flow over Airfoils

Transcription

1 Road map for Chap. 4 Incompressible Flow over Airfoils Aerodynamics 2015 fall - 1 -

2 < 4.1 Introduction > Incompressible Flow over Airfoils Incompressible flow over airfoils Prandtl (20C 초 ) Airfoil (2D) Wind (3D) Body Airfoil : any section of the wing cut by a plane normal to y-axis Aerodynamics 2015 fall - 2 -

3 < 4.2 Airfoil Nomenclature > NACA (National Advisory Committee for Aeronautics) series Thickness Mean camber line Chord line Camber Upper surface Leading edge Lower surface Trailing edge Aerodynamics 2015 fall - 3 -

4 < 4.2 Airfoil Nomenclature > NACA (National Advisory Committee for Aeronautics) series NACA 4-digit series Incompressible Flow over Airfoils * NACA : max. camber = 2% of the chord 4 : the location of max. camber = 40% of the chord 12 : max. thickness = 12% of the chord If the airfoil is symmetric, it becomes NACA00XX NACA 5-digit series * NACA : 2*0.3/2 = 0.3 design C L 30 : 30/2 % = the location of max. camber 12 : max. thickness = 12% of the chord Aerodynamics 2015 fall - 4 -

5 < 4.2 Airfoil Nomenclature > NACA (National Advisory Committee for Aeronautics) series 6-digit series laminar flow airfoil * NACA : series designation 5 : min. pressure location = 50% of the chord 2 : design C L = : max. thickness = 18% of the chord Other notations * SC0195 * VR12 Aerodynamics 2015 fall - 5 -

6 < 4.3 Airfoil Characteristics > * 1930~40 NASA carried numerous experiments on NACA airfoil characteristics (Measured C l, C d, C m 2-D data) * In the future, new airfoils should be designed and tested (consideration of aerodynamic, dynamic & acoustic limitation) * Typical lift characteristics of an airfoil Stall Sepatation Dynamic stall Maximum lift coefficient How to measure C l, C d, C m? a 0 = Zero lift angle Aerodynamics 2015 fall Stall angle (12~18deg) : angle of attack

7 < 4.3 Airfoil Characteristics > [Def.] a, angle of attack : the angle between the freestream velocity and the chord [Note] 1. a 0 is not usually a function of Re. 2. C l,max is dependent on Re. Aerodynamics 2015 fall - 7 -

8 < 4.3 Airfoil Characteristics > Typical drag & pitching moment characteristics * Aerodynamic drag = Pressure drag Sensitive to Re. (form drag) + Skin friction drag Profile drag * AC (Aerodynamic Center) [Def.] The point about which the moment is independent of AOA Subsonic : AC=c/4 Supersonic : AC=c/2 Aerodynamics 2015 fall - 8 -

9 < 4.4 Vortex Sheet > Kutta-Joukowski Theorem * Kutta (German), Joukowski(Russia) * Incompressible, inviscid flow L = r v G * G : positive clockwise Lift G G Vortex filament of strength G Aerodynamics 2015 fall - 9 -

10 < 4.4 Vortex Sheet > * g(s) = the strength of vortex sheet per unit length along s * From Biot-Savart Law * Velocity potential for vortex flow * Velocity potential at P Aerodynamics 2015 fall

11 < 4.4 Vortex Sheet > Incompressible Flow over Airfoils * Circulation around the dashed path * If (Note) The local strength of the vortex sheet is equal to the difference (jump) in tangential velocity across the vortex sheet Aerodynamics 2015 fall

12 < 4.4 Vortex Sheet > Incompressible Flow over Airfoils * Vortex Sheet - Application for inviscid, incompressible flow * Calculate g(s) to form the streamlines with a give airfoil shape (Note) Vortex sheet method is more than just a mathematical device; it also has a physical meaning ex. : Replacing the boundary layer ( ) with a vortex sheet Aerodynamics 2015 fall

13 < 4.5 The Kutta Condition > * For a circular cylinder, Incompressible Flow over Airfoils * For a given a, should have only one solution? Aerodynamics 2015 fall

14 < 4.5 The Kutta Condition > * From the experiments, we know that the velocity at the trailing-edge in finite. Kutta Condition g(te)=v 1 -V 2 =0 V(TE)=finite * The circulation around the airfoil is the value to ensure that the flow smoothly leaves the trailing edge. Aerodynamics 2015 fall

15 < 4.6 Kelvin s Circulation Theorem > * Assume) 1. Inviscid 2. Incompressible 3. No body forces Ex) Starting vortex The time rate of change of circulation around a closed curve consisting of the same fluid elements is zero [ at rest ] [ after the start ] Aerodynamics 2015 fall

16 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * Assumptions Incompressible Flow over Airfoils i) The camber line is one of the streamlines ii) Small maximum camber and thickness relative to the chord iii) Small angle of attack * Purposes i) Find g(s) ii) Use Kutta-Joukowski theorem, L =rvg Aerodynamics 2015 fall - 1 -

17 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * The component of free-stream velocity normal to the mean camber line at P From small angle assumption Aerodynamics 2015 fall - 2 -

18 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * If the airfoil is thin, : velocity normal to the camber line induced by the vortex sheet : velocity normal to the chord line induced by the vortex sheet * The velocity at point x by the elemental vortex at point x * The velocity at point x by all the elemental vortices along the chord line Aerodynamics 2015 fall - 3 -

19 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * The sum of the velocity components normal to the surface at all point along the vortex sheet is zero The fundamental equation of thin airfoil theory Aerodynamics 2015 fall - 4 -

20 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * Sysmmetric airfoil no camber, * Transform variable x into q,, Aerodynamics 2015 fall - 5 -

21 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * Check Kutta condition Indeterminant form By L Hospital s rule Aerodynamics 2015 fall - 6 -

22 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * Since we get g(q), now calculate G, L * Lift : * Lift coefficient : * Lift slope : Lift coefficient is linearly proportional to angle of attack. Aerodynamics 2015 fall - 7 -

23 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * The moment about the leading edge Aerodynamics 2015 fall - 8 -

24 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * The moment coefficient * Aerodynamic center is located at c/4 for incompressible, inviscid and symmetric airfoil (true in real world) * Center of pressure : the point at which the moment is zero Aerodynamic center : the point at which the moment is independent of aoa Aerodynamics 2015 fall - 9 -

25 < 4.8 The Cambered Airfoil > * From thin airfoil theory, Incompressible Flow over Airfoils (a) * For cambered airfoil, Transform x into q (b) * The solution becomes (c) Leading term for symmetric airfoil Fourier series term due to camber Aerodynamics 2015 fall

26 < 4.8 The Cambered Airfoil > * Substitute (c) into (b) Incompressible Flow over Airfoils By using the integral standard form Aerodynamics 2015 fall

27 < 4.8 The Cambered Airfoil > For Fourier cosine series, Incompressible Flow over Airfoils [Note] given Determine g(q) to make the camber line a streamline with A 0, A n + Kutta condition, g(p)=0 Aerodynamics 2015 fall

28 < 4.8 The Cambered Airfoil > * The total circulation due to the entire vortex sheet By using, Aerodynamics 2015 fall

29 < 4.8 The Cambered Airfoil > * Lift coefficient for a cambered thin airfoil Lift slope, Aerodynamics 2015 fall

30 < 4.8 The Cambered Airfoil > * Lift coefficient for a cambered thin airfoil [Note] * Lift slope is 2p for any shape airfoil * Zero lift angle : Aerodynamics 2015 fall

31 < 4.8 The Cambered Airfoil > * The total moment about the leading edge * Moment coefficient Aerodynamics 2015 fall A 1 & A 2 both are independent of aoa The quarter-chord is the aerodynamic center for a cambered airfoil

32 < 4.8 The Cambered Airfoil > * The center of pressure Incompressible Flow over Airfoils Not a convenient point Aerodynamics 2015 fall

33 < 4.8 The Cambered Airfoil > The influence of camber on the thin airfoil * The cambered airfoil * The symmetric airfoil Aerodynamics 2015 fall

34 < 4.10 The Vortex Panel Method > * Thin airfoil theory - Closed form - Limited to thin airfoil, * Panel method - Vortex panel - Source panel non-lifting cases * Exactly same idea of thin airfoil theory, but no closed form g(s) solve numerically Aerodynamics 2015 fall - 1 -

35 < 4.10 The Vortex Panel Method > Boundary point J-1 j y q j x P(x,y) Control point (x j,y j ) J+1 * The velocity potential at P due to j-th panel * Let s put point P at the control point of i-th panel Aerodynamics 2015 fall - 2 -

36 < 4.10 The Vortex Panel Method > * At the control points, the normal component of velocity is zero. - The component of V normal to i-th panel - The normal component of induced velocity at (x i, y i ) = : f (panel geometry) Aerodynamics 2015 fall - 3 -

37 < 4.10 The Vortex Panel Method > * Boundary condition : + Kutta condition : i i+1 * Now, we have (n+1) eq. with n unknowns ignore one of control points * The flow velocity tangent to the surface = g u i,1 u i,2 * Total circulation : * Lift : Inside the solid surface Aerodynamics 2015 fall - 4 -

38 < 4.12 The Flow over an Airfoil the Real Case > Stall Leading-edge stall Flow separation takes place over the entire top surface of the airfoil after occurring at the leading edge Aerodynamics 2015 fall - 5 -

39 < 4.12 The Flow over an Airfoil the Real Case > Stall Trailing-edge stall α = 5 α = 10 α = 15 α =22.5 Flow separation takes place from the trailing edge at thicker airfoils than leading-edge stall Aerodynamics 2015 fall - 6 -

40 < 4.12 The Flow over an Airfoil the Real Case > Stall Thin airfoil stall Leading-edge stall Flow separation takes place over the entire surface of the airfoil after occurring at the leading edge Aerodynamics 2015 fall - 7 -

41 Lift coefficient Incompressible Flow over Airfoils < 4.12 The Flow over an Airfoil the Real Case > Stall Lift-coefficient curves 1.5 Leading-edge stall 1.0 Trailing-edge stall 0.5 Thin airfoil stall α, degrees Aerodynamics 2015 fall - 8 -

42 < 4.12 The Flow over an Airfoil the Real Case > High-lift devices Leading edge slat Trailing edge flap Aerodynamics 2015 fall - 9 -

43 < 4.12 The Flow over an Airfoil the Real Case > High-lift devices Trailing-edge flap (plain type) More camber Higher lift Aerodynamics 2015 fall

44 < 4.12 The Flow over an Airfoil the Real Case > High-lift devices Effect of slats and flaps Aerodynamics 2015 fall