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National Advisory Committee for Aeronautics, Report - Airfoil Profiles for Minimum Pressure Drag at Supersonic Velocities - General Analysis with Application to Linearized Supersonic Flow

A theoretical investigation is made of the airfoil profile for
minimum pressure drag at zero lift in supersonicfloie. In the
first part of the report a general method is developed for calcu-
lating the pmfile having the least pressure drag for a giren
auriliary condition, such as a giren structural requirement or a
giren thickness ratio. The carious structural requirements con—
sidered include bending strength, bending stifiness, torsional
strength. and torsional stillness. No assumption is made
regarding the trailing-edge thickness; the optimum value is
determined in the calculations as afunction of the base pressure.

To illustrate the general method, the optimum airfoil, defined
as the airfoil having minimum pressure drag for a giren auxil—
iary condition, is calculated in a second part of the report using
the equations of linearized supersonicfioun It isfound that the
optimum airfoil in most cases has a blunt trailing edge. I t also
'isfound that the optimum thickness distribution depends only on
one dimensionless parameter, termed the “base pressure param-
eter”. This parameter inrolres the Mach number, airfoil
thickness ratio, and base pressure coefiicient. The eject of
variations in each of these latter three quantities on the shape of
the optimum profile is discussed, and a simple criterion formu-
lated for determining the condition under which the optimum
trailing-edge thickness is greater than zero. The calculated
pressure drag of the optimum profile is compared to that of a.
b-iconcer sharp—trailing-edge profile satisfying the same struc-
tural requirement. The reduction in pressure drag depends on
the base pressure parameter and rariesfrom a few percent to as
much as to” percent.

In supersonic flow the finite thickness of an airfoil invari—
ably introduces a certain amount of pressure drag which can
be minimized by a rational choice of airfoil shape. The
profile for minimum pressure drag depends, among other
things, on the particular auxiliary condition that is imposed
on the airfoil geometry. For example, if it is required that
the optimum profile (defined herein as the profile of least
pressure drag for a given aum'liary condition} satisfy the
auxiliary condition of a given thickness ratio, then according
to a well—known result of Ackeret’s linearized airfoil theory,
the so-callcd double-wedge profile represents the optimum
sharp—trailing-edge airfoil.