# 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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