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National Advisory Committee for Aeronautics, Technical Notes - Some Conical and Quasi Conical Flows in Linearized Supersonic Wing Theory

A number of conical and quasi-conical linearized supersonic
flows have been derived° These flows may be applied in lift-
cancellation techniques in the deterndnation of wing-lift dis-
turbances that arise at subsonic trailing edges. Some of the
results are applied to damping in roll and pitch in another paper.

Two methods of analysis have been employed, both involving
integral equations. One is a development of the membrane method
of Evvard; the other is an analog of the superposition method of
Schlichting. In both cases, source distributions are used. The
mathematical equivalence of the first method and a new doublet-
distributicn method of Goodman and Mirels is shown.

The analysis of the flow over a sweptback wing (reference 1)
makes use of Lagerstrom’s concept of lift cancellation (reference 2).
The starting point is a delta wing of infinite chord. Suitable flows
are superimposed that cancel the lift outboard of and behind a '
certain boundary. This boundary is/chosen to constitute the tips
and the trailing edge of the sweptback Wing.

The cancellation flows for the wing tip modify the lift in the
tip region; they are not of concern herein. The cancellation flows
for the trailing edge modify the lift in a region ahead of the trail-
ing edge if the edge is "subsonic"(that is, if the component stream
velocity normal to the edge is subsonic). Now the lift distribution
of a delta wing at an angle of attack is substantially flat in the
center. (See fig. 1.) Thus, an approximate cancellation behind the
trailing edge would be afforded by a constant lift such as flow I in
the figure. (This idea was originated in reference 1.) Correspond-
ing approximate cancellation flows (III and IV) for rolling and
pitching motions are also shown in the figure. The derivation of
these three flows is the main Object of this report; the work was
initiated at the NAGA.Langley laboratory and completed at the NACA
Lewis laboratory.