2-D Inviscid, Linearized, Thin Airfoil Theories

14.7.1.1 Incompressible Flow (Mo = 0)

Cambered Plate Geometry

Consider a thin cambered plate of chord c of equation

d(x) = AcX (1 – 2X) (1 – X)

Calculate d'(x) to help you with the graph and make a plot of the plate (Hint: expand d (x) before taking the derivative, A is not determined at this point). Verify your result as much of the rest depends on it.

Fourier Coefficients

Find all the Fourier coefficients Ao, A1; A2,An for this thin cambered plate and give the incidence of adaptation aadapt. Use the identity cos21 = 2 (1 + cos 2t).

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), Cm, o (a) and Cm, a.c.

Static Equilibrium About an Axis

If an axis is located at the mid-chord, f = 5, find the equilibrium angle aeq if there are no forces other than the aerodynamic forces. Is the equilibrium stable, unstable, neutral?

14.7.1.2 Supersonic Flow (M0 > 1, в = JM( — 1)

The same cambered plate equips the fins of a missile 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).

Static Equilibrium About an Axis

If an axis is located at the mid-chord, f = 2, find the equilibrium angle aeq if there are no forces other than the aerodynamic forces. Is the equilibrium stable, unstable, neutral? (Hint: calculate Cm 1).

m, 2

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