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National Advisory Committee for Aeronautics, Technical Notes - Comparison of Supersonic Minimum Drag Airfoils Determined by Linear and Nonlinear Theory

Supersonic profiles of minimum pressure drag for a given thickness
ratio and for a given area have been determined with the use of a non—
linear pressure relation and are compared with minimum—drag profiles
found by linearized theory. The results show that the profiles are
determined with sufficient accuracy by linear theory over the entire
supersonic Mach number range since the drag coefficients for these
profiles are only slightly higher than those fer optimum profiles deter-
mined by nonlinear theory. Linear theory appears to be adequate for
determining profiles of minimum drag for other auxiliary structural
conditions since moderate deviations from the optimum shape have only
a small influence on the pressure drag.

The parameters determining the airfoil shape for a given thickness
ratio found by both the linear and nonlinear theory are presented in
graphs as a function of the base pressure coefficient. With the use of
these results, the optimum profiles for any stream Mach number and thick—
ness coefficient are readily determined. A comparison of the pressure
drag coefficients for optimum profiles determined by linear and nonlinear
theory is presented for the Mach number range from 1.5 to 10.0. In
addition, several optimum profiles for a given area have been calculated
by both the linear and nonlinear theory.

Drougge (reference 1) has determined the airfoil section shape for
minimum pressure drag at supersonic speeds subject to such auxiliary
conditions as given bending and torsional stiffness. These calculations
were made by using the linearized expression for the pressure coefficient;
the effect of a base was not considered. Recently, Chapman (reference 2)
has shown that the section shape for minimum pressure drag as determined
by linearized theory may have a blunt trailing edge. ,The use of linear-
ized theory for determining optimum profiles facilitates the mathematical
development; however, the results are subject to question particularLy
at high-supersonic Mach,numbers.

The purpose of the present paper is to compare the section shapes
for minimum pressure drag (subject to certain auxiliary conditions)
determined by linear and nonlinear theory in order to estimate the
errors introduced by the linearized form of the pressure coefficient
and to determine its range of validity for calculations of this nature.
For this purpose, it was considered sufficient to examine two prdblems.
The problems chosen were the determination of the profile for minimum
drag for a given thickness ratio and the determination of the profile
for minimum drag for a given area.