Thin Airfoil Theory Derivation


We start with the analysis of a very thin cambered plate and will build up the solution to a more arbitrary airfoil.

A distribution of vorticity on the airfoil will be a solution to Laplace's equation. It will satisfy the boundary conditions if the combination of the velocity induced by the vortices cancels the component of the freestream normal to the plate:

(where small angle approximations have been introduced)

The basic approximation of thin airfoil theory is that the velocity induced
at some point x due to the vorticity at x'...

... may be approximated by the velocity induced at the same x position on the x axis due to a vortex on the x axis:

The velocity induced by this bit of vorticity is computed from the basic vortex singularity. The formula is known as the Biot-Savart Law and in 2-D for the element of vorticity at x', it reads:



So, the total induced velocity at the point x is given by:

Combining this expression with the flow-tangency boundary condition, we have the basic integral equation to be solved for the unknown vorticity distribution:



The approach to solving this equation is to change variables:


and to write γ as a Fourier series:

Substituting (4) into (2) and using the trigonometric relations:

yields:

Finally, we multiply both sides by cos m θ and integrate from 0 to π:


We can substitute these coefficients into the expression for γ (4), to find the vorticity, pressures, lift, and moment as a function of the surface slope distribution, dz/dx.

In particular, the expressions for the local pressure difference and integrated lift and moment about the leading edge are:

(Note that the expression relating Cp and γ applies only to thin airfoils. )

This method of computing the circulation distribution permits us to compute the pressure difference across the airfoil. The pressures on the upper and lower surfaces may also be computed by noting that at the surface the perturbation velocities in the x direction caused by the singularities are zero except for those due to the local vorticity:

The only contribution to du(x) is from the local vorticity (at x). It can be shown that this perturbation velocity is du = ± γ/ 2, with the + sign for the upper surface and the - sign for the lower surface.