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National Advisory Committee for Aeronautics, Technical Notes - Theoretical Study of the Transonic Lift of a Double Wedge Profile with Detached Bow Wave

A theoretical study is described of the aerodynamic characteristics
at small angle of attack of a thin, double-wedge profile in the range
of supersonic flight speed in which the bow wave is detached. The
analysis is carried out within the framework of the transonic (nonlinear)
small- disturbance theory, and the effects of angle of attack are regarded
as a small perturbation on the flow previously calculated at Zero angle.
The mixed flow about the front half of the profile is calculated by
relaxation solution of a suitably defined boundary-value problem for the
transonic small—disturbance equation in the hodograph plane (i.e., the
Tricomi equation). The purely supersonic flow about the rear half is
found by an extension of the usual numerical method of characteristics? /
Analytical results are also obtained, within the framework of the same
theory, for the range of speed in which the bOW'wave is attached and the
flow is completely supersonic.

The calculations provide, for vanishingly small angle of attack,
the following information as a function of the transonic similarity
parameter: (1) chordwise lift distribution, (2) lift-curve slope, and
(3) position of center of lift. As in previous studies, the aerodynamic
characteristics of a profile of given thickness ratio show little varia-
tion with free—stream Mach number as the Mach number passes through 1.
As the Mach number is increased to higher values, however, the lift—
curve slope rises to a pronounced maximum in the vicinity of shock
attachment and then declines. Correspondingly, the center of lift moves
forward toward the leading edge and then returns aft. These findings
are in marked contrast to the behavior of the drag coefficient at zero
angle of attack, which was found in earlier work to decrease monotoni-
cally as the Mach number increased above 1. At Mach numbers above that
for shock attachment, the results of the present calculations tend toward
those given by 'classical linear theory.

The theoretical problem of the transonic flow over a thin, double—
wedge profile at zero angle of attack has been treated in several papers
in recent years. These papers have in common that they employ the
simplifying concepts of the transonic small-disturbance theory and
utilize the hodograph transformation to linearize the resulting mathe—
matical problem. Following this approach, Guderley and Ybshihara
(reference 1) began by solving the problem for a free—stream Mach
number of 1, using analytical methods for the mixed flow over the front
wedge and the method of characteristics for the purely supersonic flow
over the rear. Somewhat later, the present authors, using a combina—
tion of relaxation methods and the method of characteristics (refer—
ences 2 and 3), extended the results to free-stream Mach numbers greater
than 1, where a detadhed bow wave occurs ahead of the profile.