# 2-D Inviscid, Linearized, Thin Airfoil Theories

15.10.1.1 Incompressible Flow (M0 = 0)

Profile Camber Estimation

Thin parabolic plate has only two non-zero Fourier coefficients:

, , . dm

A0 = a, A1 = 4—

c

In thin airfoil theory, the lift coefficient is given by

Q = 2^ A0 + Afj

Solving for A1, given Ci and a reads

( Ci (2.1878

A1 = 2 1 — – ^ = 21 ——————- 0.0698П = 0.5568

1 2ъ ) 6.2832 )

The Fourier coefficients are

A0 = a, A1 = 0.5568, A2 = A3 = ••• = An = 0, n > 2 The relative camber dm/c of the thin parabolic plate is A1/4

= 0.14

c

a 14% relative camber!

Take-Off Incidence

For a lift coefficient C; = 2.5, the incidence needed is

Nose Pitching Moment Coefficient

In thin airfoil theory, the pitching moment coefficient Cm, o (a) is given by

The nose pitching moment coefficient of the wing profile (Cm,0)profile at a = 4° is predicted to be

15.10.1.2 Supersonic Flow (Mq > 1, в = ^

Consider the parabolic plate of equation

x

d (x) = 4dm — c

The slope is given by

d

where the relative camber is given to be dm/c = 0.14.

Lift Coefficient

The lift coefficient Ci (a) for all thin airfoils is

a

Ci (a) = 4­P

Drag Coefficient

The equilibrium incidence corresponds to Cm, c (a) = 0, which gives aeq = -0.3733 rd =-21.4°. ’4

The equilibrium is stable, since the moment slope is negative.