Airfoil - Derivation of Thin Airfoil Theory

Derivation of Thin Airfoil Theory

The airfoil is modeled as a thin lifting mean-line (camber line). The mean-line, y(x), is considered to produce a distribution of vorticity along the line, s. By the Kutta condition, the vorticity is zero at the trailing edge. Since the airfoil is thin, x (chord position) can be used instead of s, and all angles can be approximated as small.

From the Biot-Savart law, this vorticity produces a flow field where

where is the location where induced velocity is produced, is the location of the vortex element producing the velocity and is the chord length of the airfoil.

Since there is no flow normal to the curved surface of the airfoil, balances that from the component of main flow, which is locally normal to the plate—the main flow is locally inclined to the plate by an angle . That is:

This integral equation can by solved for, after replacing x by

,

as a Fourier series in with a modified lead term

That is

(These terms are known as the Glauert integral).

The coefficients are given by

and

By the Kutta–Joukowski theorem, the total lift force F is proportional to

and its moment M about the leading edge to

The calculated Lift coefficient depends only on the first two terms of the Fourier series, as

The moment M about the leading edge depends only on and, as

The moment about the 1/4 chord point will thus be,

.

From this it follows that the center of pressure is aft of the 'quarter-chord' point 0.25 c, by

The aerodynamic center, AC, is at the quarter-chord point. The AC is where the pitching moment M' does not vary with angle of attack, i.e.,