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National Advisory Committee for Aeronautics, Technical Notes - Corrections for Lift, Drag, and Moment of an Airfoil in a Supersonic Tunnel Having a Given Static Pressure Gradient

The corrections for lift, drag, and moment of a two-dimensional air-
foil have been analyzed, on:the assumption that the airfoil is tested in
the working section of a supersonic tunnel in which the pressure field,
instead of being uniform, is characterized by gradients in the axial and
transverse directions.

The pressure gradients of the tunnel as well as the effect of the
airfoil to be tested are regarded as perturbations of the original rec-
tilinear flow field of given Mach number. Therefore the velocity poten—
tial of the flow, the nonlinear differential equation of motion, and the
boundary conditions are expanded into double series in powers of two
parameters, one characterizing the airfoil thickness 6 and the other,
the inhomogeneity of the field b. In this way the nonlinear problem
is split into a system of linear boundary-value problems, corresponding
to the different powers of b and e.

In each of the resulting problems there appears, besides the dif—
ferential equation and boundary condition, an additional condition to
be stipulated on the characteristics passing through the leading edge.
Particular attention has been paid to the correct formulation of this
"characteristic condition."

The solution procedure is carried out up to orders b2, 62, and
as in the velocity potential. This means that, for example, the drag
is computed up to orders b26, 63, and 62b. The physical meaning of
the results is discussed. The drag term in eb represents the
"horizontal buoyancy" of the airfoil, the term proportional to be2 is
a consequence of the interaction of the airfoil field and the inhomo-
geneous pressure field, and the term in b26 may be considered as a
"second-order buoyancy." The meaning of the various lift and moment
terms may be interpreted similarly. The resulting expressions have been
derived for arbitrary given pressure gradients and general profile form.

All solutions are obtained in closed, analytic form ready for imme-
diate evaluation. Representative examples with graphs are included.