Incompressible Flow over Airfoils

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1 < 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 2017 fall - 1 -

2 < 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 2017 fall - 2 -

3 < 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 2017 fall - 3 -

4 < 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 2017 fall - 4 -

5 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * Sysmmetric airfoil no camber, * Transform variable x into q,, Aerodynamics 2017 fall - 5 -

6 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * Check Kutta condition Indeterminant form By L Hospital s rule Aerodynamics 2017 fall - 6 -

7 < 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 2017 fall - 7 -

8 < 4.7 Classical Thin Airfoil Theory > The Symmetric Airfoil * The moment about the leading edge Aerodynamics 2017 fall - 8 -

9 < 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 2017 fall - 9 -

10 < 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 2017 fall

11 < 4.8 The Cambered Airfoil > * Substitute (c) into (b) Incompressible Flow over Airfoils By using the integral standard form Aerodynamics 2017 fall

12 < 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 2017 fall

13 < 4.8 The Cambered Airfoil > * The total circulation due to the entire vortex sheet By using, Aerodynamics 2017 fall

14 < 4.8 The Cambered Airfoil > * Lift coefficient for a cambered thin airfoil Lift slope, Aerodynamics 2017 fall

15 < 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 2017 fall

16 < 4.8 The Cambered Airfoil > * The total moment about the leading edge * Moment coefficient Aerodynamics 2017 fall A 1 & A 2 both are independent of aoa The quarter-chord is the aerodynamic center for a cambered airfoil

17 < 4.8 The Cambered Airfoil > * The center of pressure Incompressible Flow over Airfoils Not a convenient point Aerodynamics 2017 fall

18 < 4.8 The Cambered Airfoil > The influence of camber on the thin airfoil * The cambered airfoil * The symmetric airfoil Aerodynamics 2017 fall

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