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National Advisory Committee for Aeronautics, Technical Notes - Determination of Indicial Lift and Moment of a Two Dimensional Pitching Airfoil at Subsonic Mach Numbers from Oscillatory Coefficients with Numerical Calculations for a Mach Number of 0.7

The reciprocal equations relating the lift and moment acting on an
airfoil due to harmonic oscillations in compressible flow to the indicial
lift and moment functions for a pitching airfoil are determined. Two
indicial functions are required to describe completely the lift or
moment on the airfoil; one function is used for describing the effects
due to the angular position of the airfoil whereas the second function
is used for describing the effects due to pitching velocity.

Calculations for the indicial lift and moment functions are made
for a pitching airfoil at a Mach number of 0.7 for which sufficient
oscillatory coefficients were available. The growth of lift to the
steady-state value is less rapid for compressible flow than for incom—
pressible flow for both indicial functions of the pitching airfoil.
Although the circulatory lift due to pitching velocity on an airfoil
rotating about its three-quarter-chord point is nonexistent at a Mach
number of O, for a Mach number of 0.7 a time—dependent lift exists
because of pitching velocity and approaches zero in approximately
2 chords.

In the study of transient flows, two types of airfoil motions have
special significance — a harmonically oscillating airfoil and an air-
foil experiencing a sudden change in angle of attack. The reciprocal
relations between the indicial lift and the lift associated with a
harmonically oscillating airfoil for both sinking and pitching motion
are given in reference 1 for incompressible flow. In reference 2 the
reciprocal relations were extended to apply to an airfoil suddenly
acquiring a vertical velocity in a subsonic compressible flow, and the
indicial lift and.moment functions were evaluated for a Mach number
of 0.7. The present paper is an extension of reference 2 in that the
indicial lift and moment functions for a pitching airfoil are determined.
The reciprocal equations for the case of a pitching airfoil are indi-
cated, and the indicial lift and moment functions are computed for a
Mach number of 0.7.

In a recent paper by Lomax, Heaslet, and Sluder (reference 3), a
different method for determining the indicial lift and moment functions
is given. Although parts of the indicial functions can be determined
readily, the solution for the complete indicial functions is lengthy
and tedious, and, consequentby, numerical results are given only for a
Mach number of 0.8 in reference 3. Part of the solution presented herein
for a Mach number of 0.7, however, was determined by the method in refer-
ence 3 and was compared with the solution determined in this paper.