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National Advisory Committee for Aeronautics, Report - Theoretical Prediction of Pressure Distributions on Nonlifting Airfoils at High Subsonic Speeds

Theoretical pressure distributions on nonlifting circular—arc
airfoils in two-dimensionalflows with high subsonic free-stream
velocity are found by determining approximate solutions, through
an iteration process, of an integral equation for transonic flow
proposed by Oswatitsch. The integral equation stems directly
from the small-disturbance theory for transonic flow. This
method of analysis possesses the advantage of remaining in the
physical, rather than the hodograph, variables and can be applied
to airfbils having curved surfaces. After discussion of the derive»
tion of the integral equation and qualitative aspects of the solu-
tion, results of calculations carried out for circular-arc airfoils in
flows with free-stream Mach numbers up to unity are described.
These results indicate most of the principal phenomena observed
in experimental studies. At subcritical Mach numbers, the pres-
sure distribution is symmetrical about the midch position and
the drag is zero. The magnitude of the pressure coefiicient is
found to increase more rapidly with increasing Mach number
than the Prandtl—Glauert rule would indicate. When the critical
Mach number is exceeded, compression shocks occur, the fore—
and—aft symmetry of the pressure distribution is lost, and the air-
foil experiences a drag force. As the Mach number is increased
further, the shock wave becomes of greater intensity and moves
rearward along the chord, thereby producing a rapid increase in
the magnitude of the pressure drag coeflicient. At Mach num-
bers close to unity, the variation of the pressure, local Mach
number, and drag conforms, within the limitations of transonic
small perturbation theory, to the known trends associated with
the Mach number freeze. Some comparisons with experimental
results are also included.

The solutions are obtained using an iteration process which
diflers from the classical methods in that the quadratic nature
of the integral equation is recognized. If the iteration calcula-
tions are started using the linear—theory solution, it is shown that
the retention of the quadratic feature has the interesting effwt of
forbidding shock-free supercritical second—order solutions. In
order to obtain solutions for supercritical Mach numbers, it is
necessary to start the iteration calculations with a velocity or
pressure distribution which contains a compression shock. When
this is done, it is found that the iteration procedure converges to
a definite result.