# 2-D Inviscid, Linearized, Thin Airfoil Theories

14.9.1.1 Incompressible Flow (M0 = 0)

Aerodynamic Center

Give the definition of the aerodynamic center.

Where is the aerodynamic center located for thin airfoils at low speeds?

Second Mode only Airfoil

Consider a thin cambered plate such that the vorticity distribution is given by the second mode with A2 > 0 as

/ Г'[x(t)]=2U {A01+ft + A2 cos 2t}

x (t) = 2 (1 – cos t), 0 < t < n

Use the formula for d'[x(t)] to find A0 and find the incidence of adaptation for this airfoil (Hint: use the identity cos 2t = 2 cos21 – 1, and integrate in x: d'(x)dx from zero to c; or in t: d'[x(t)]dxdt from zero to n.)

Eliminate A0 and sketch the slope of the cambered plate along with the profile itself.

Aerodynamic Coefficients

Give the expression of C;(a), Cm, o(a), Cm, a.c. and Cd for this airfoil.

Moment About an Axis

Calculate the aerodynamic moment about the mid-chord, Cm, c/2 (Hint: use the change of moment formula.)

If the profile is allowed to rotate without friction about an axis located at midchord, find the equilibrium incidence, aeq, if only aerodynamic forces and moment are present.

Is the equilibrium stable (Answer by Yes or No)?

14.9.1.2 Supersonic Flow (M0 > 1, в = ^M^ — 1)

Consider the cubic plate of equation

4 x x x

d (x) = – Ac 1 – 2 1 – , A > 0

3 c V c c)

The slope is given by

. 4 ( x x2

d (x) = 3 A 1 — 6 —+ 6 —2

This plate equips the fins of a supersonic rocket.

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).

Moment Coefficient

At a = 0 , calculate the moment coefficient (Cm,0) 0 and give the expression of

Cm, o(a).

Use the change of moment formula to evaluate Cm, c/2 and discuss whether or not there is an equilibrium about a mid-chord axis and why. How would you qualify this situation: stable, unstable, neutral?

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