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National Advisory Committee for Aeronautics, Technical Notes - Theoretical Pressure Distributions for Several Related Nonlifting Airfoils at High Subsonic Speeds

Theoretical pressure distributions on five related airfoils,
including thin symmetrical circular-arc airfoils, in two-dimensional flows
with high subsonic free-stream velocity are presented. The airfoils have
various locations for the point of maximum thickness ranging from 30— to
"(O-percent chord and are of arbitrary, although small, thickness ratio.
The results are obtained by approximate solution, through an iteration
process, of a nonlinear integral equation derived from the equations of
transonic flow theory. It is shown that the pressure distributions dis-
play most of the principal phenomena observed in experimental studies,
and are in good correspondence with those calculated by other methods for
subcritical Mach numbers and for Mach numbers near 1.

The equations governing transonic flows are known and well established
by favorable comparisons with experiment. Although the difficulties aris-
ing as a result of the nonlinearity and mixed character of the differential
equation for the potential have hindered the advancement of the analysis,
approximate methods are gradually emerging that permit the theoretical
prediction of pressure distributions on a wide variety of shapes of
aerodynamic interest.

One of these methods is that described in reference 1 in which the
differential equation of transonic flow theory is recast into the form
of a nonlinear integral equation and approximate solutions are sought by
application of an iteration procedure. The iteration procedure is of the
successive approximation type, but differs from the related procedures
customarily emloyed to detennine higher approximations to the solutions
of problems of compressible flow theory in that the quadratic nature of
the integral equation is recognized and retained throughout the analysis.
This method is described in general in reference 1 and is applied to a
nonlifting symmetrical circular-arc airfoil for which pressure distribu-
tions are calculated for a range of Mach nmnbers extending from well below
the critical Mach number up to unity.