Solution for Prescribed Freestream Fluctuations

Airfoil encountering a transverse gust
Assume that the gust velocity is a unit step function of magnitude w0 which is located at the leading edge at t=0, and is convected at the freestream speed.  where  is the unit step function.
Gust velocity at midchord becomes:
. Its Fourier transform is:

To obtain this, set  ; apply the Fourier transform to the normalwash at midchord:  .

To use this, we must first develop an expression for cl for a case of simple harmonic gust velocity variation, then substitute the indicial gust expression n the frequency domain. As before, we seek a generalized expression where the complicated parts of the integral are computed as a general function.

Simple harmonic transverse gust:

In flow-fixed coordinates,  , a simple harmonic spatial function. In terms of  , non-dimensional body-fixed coordinates,  ; periodic in space and time.

Substitute in our expression for Cp in the frequency domain. Now

and  . Substituting for Cp, we get (skipping some math here),  , and  .

Define . Thus . Transform cl(k,t) into the frequency domain:  and

and  . Substituting for  , with  (semi-chords traveled in time t),


is the Kussner Function. 

Approximate Expressions:
, or  , both appropriate near s=0, and
, which is more appropriate away from s=0.
In the figure below, the Wagner and Kussner functionsare plotted as functions of s.