2-D Inviscid, Linearized, Thin Airfoil Theories

14.8.1.1 Incompressible Flow (M0 = 0)

Profile Geometry

Подпись: f + (x) = d(x) + 2e(x) = ©x, 0 < x < 2 f + (x) = d(x) + 1 e(x) = ©(c — x), 2 < x < c 2-D Inviscid, Linearized, Thin Airfoil Theories

Consider a half-double wedge profile of chord c of equation

where © is the wedge angle, see Fig. 14.3.

Calculate the distributions of camber d(x) and thickness e(x) for this profile. Check your result.

Fig. 14.3 Half double-wedge geometry

Fourier Coefficients

Give the expressions of the Fourier coefficients A0 and An in the expansion of the vorticity for an arbitrary profile.

Calculate the Fourier coefficients A0, A1 and A2 for this thin profile (Hint: you

c

need to split the integral into two pieces J02 + Jc).

Give the incidence of adaptation aadapt.

Sketch the flow at the incidence of adaptation, showing in particular the stream­lines near the leading and trailing edges.

Definition of Aerodynamic Center

Give the definition of the aerodynamic center.

Aerodynamic Coefficients

Give the aerodynamic coefficients C;(a) and Cm, o (a).

14.8.1.2 Supersonic Flow (M0 > 1, в = yjM( — 1)

The same profile equips the wing of an airplane cruising at Mach number M0 > 1 in a uniform atmosphere.

Pressure Distribution and Flow Features

Calculate and plot —C + and – C— versus x for this airfoil at a = 0. Sketch the flow at a = 0 (shocks, characteristic lines, expansion shocks).

Aerodynamic Coefficients

At a = 0 , calculate the drag and moment coefficients (Cd)a=0 and (Cm, o) 0

Static Equilibrium About an Axis

If an axis is located at the leading edge, x = 0, find the equilibrium angle aeq if there are no forces other than the aerodynamic forces. Is the equilibrium stable, unstable, neutral? (Hint: calculate Cm, o (a)).

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