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National Advisory Committee for Aeronautics, Report - Calculations on the Forces and Moments for an Oscillating Wing Aileron Combination in Two Dimensional Potential Flow at Sonic Speed

The linearized theory for compressible unsteady flow is used,
as suggested in recent contributions to the subject, to obtain the
velocity potential and the lift and moment for a thin, harmoni-
cally oscillating, two-dimensional wing~aileron combination
moving at sonic speed. The velocity potential is derived by
considering the sonic case as the limit of the linearized super-
sonic theory. From the velocity potential acplicit expressions
for the lift and moment are developed for vertical translation and
pitching of the wing and rotation of the aileron. The report
provides ean‘ensive tables of numerical values for the coefiicients
contained in the expressions for lift and moment, for various
values of the reduced frequency k (0<k§3.5), and aileron
hinge position (from 10 to .90 percent of the wing chord). The
sonic results are compared and found to be consistent with
previously obtained subsonic and supersonic results. Several
figures are presented showing the variation of lift and moment
with reduced frequency and Mach number and the influence of
Mach number on some cases of bending-torsion flutter.

Instability investigations for high-speed aircraft often
require a knowledge of the air forces and moments that act
on an oscillating wing moving at high speed. For subsonic
and supersonic speeds the main source of theoretical in-
formation has been the solution of the linearized differential
equation for compressible flow. For sonic or near-sonic
speed, however, the linearized theory has been generally
assumed inapplicable, since it does not allow for thickness
effects, shocks, and strong disturbances As is well known,
it predicts infinite forces on a nonoscillating, thin, unswept
wing moving at sonic speed.

Important differences exist, however, between the steady
and unsteady cases By a discussion of the order of magni-
tude of the terms of the general nonlinear differential equa-
tion for compressible flow, reference 1 shows that for unsteady
two—dimensional flow at sonic speed this equation 'is essen-
tially linear and in linear form leads to physically plausible
results for the forces on a thin oscillating wing, provided the
frequency of oscillation is sufficiently large; A similar con-
clusion was reached in reference 2, where linear methods
applied to a wingin two~dimensional nonstationary flow at
sonic speed yielded perturbation velocities of the same order
of magnitude as those obtained for subsonic or supersonic
speeds.