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National Advisory Committee for Aeronautics, Technical Notes - The Twisted Wing with Elliptic Plan Form

The chief numerical results referring to the aerodynamic induc—
tion of single wings are derived from the investigation of wings
with elliptic plan views with the angle of attack constant all over
the span. It was found from wind tunnel tests (reference (1) ) and
later confirmed by means of a very laborious computation by A. Betz
heference (5) ) that a wing with rectangular plan form has practical—
ly the same average induction as the elliptic wing; The plan form
of actual wings is in general neither rectangular nor elliptic but
something between these two, and'it is often better described by com-
paring it with an ellipse than by comparing it with a rectangle.

The results obtained for the elliptic wing are thus even more useful
for actual wings than for rectangular wings. It seems therefore inu
structive and interesting to explore the elliptic wing farther and to
investigate more general elliptic wings, no longer subjected to the
condition of constant angle of attack. I will assume in this paper
the plan view to be elliptic but the angle of attack to be variable
and to be different at different distances from the center. I con-
sider first cases where the angle of attack is some special function
0f the Span b and proceed afterwards to the most general case where
any twist or any distribution of the lift is given. The results can
be applied beyond the original conception of a twisted elliptic wing.
Some other-problems can be treated by the consideration of wings with
equivalent twist. These applications however are still under study
and will be presented later separately.

The elliptic wing with the twist zero, that is, with constant
geometric angle_of attack is distinguished by an extremely'simpge re—
lation between the effective angle of attack and the induced angle of
attack. Under the usual assumptions (reference (3) ) these two can
be described by the same function of the span; the ratio of the two
angles is constant all over the span. Hence this is also true for
their sum, the geometric angle of attack, and all three angles are
constant. The relation between the three angles is linear and inde—
pendent of the distance from the center and this characteristic rathp

er than the fact that the three angles themselves are constant makes
the results so simple and useful. Therefore the Question suggests
itself whether there cannot be found other variations of the angle
of attack for which the same relation holds true, viz., that the cf;
fective angle of attack and the induced angle of attack are express—
ed by the same function.with a different constant factor, which func-
tion then also expresses the geometric angle of attack. It is not
easy to arrive in a systematic way at the solution_of this problem._
It must be enough to describe in the next paragraph the solutions '
.and to demonstrate that they really conform to the condition.