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?