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National Advisory Committee for Aeronautics, Technical Notes - Corrections for Drag, Lift, and Moment of Axially Symmetrical Body Placed in a Supersonic Tunnel Having a Two Dimensional Pressure Gradient

The corrections for drag, lift, and moment are derived for an
axially symmetrical body placed in the test section of a supersonic tun-
nel, on the assumption that the test section is characterized by a two-
dimensional pressure field originating from construction flaws. Although
relatively simple longitudinal and transverse pressure gradients are
assumed, the analytical treatment becomes rather difficult because of the
difference in symmetry between the body and the basic flow field.

Assuming irrotational conditions, the velocity potential of the flow
around the body is expanded in a threefold manner: (1) In powers of the
thickness parameter of the body 6, (2) in powers of a parameter b
characterizing the inhomogeneity of the basic flow field, and (3) as a
Fourier series in the azimuth 6 around the body axis. Each expansion
is taken into account not further than up to the second term.

Upon substitution of this potential series, the nonlinear equation
of motion and the boundary condition on the body surface are split into
a set of linearized boundaryevalue problems which can be solved analyti-
cally. The mathematical techniques used for the solution are explained
in appendixes.

Assuming the two—dimensional pressure field, the drag, lift, and
moment corrections for arbitrary body shapes are obtained in closed ana-
lytic form. The physical meaning of the results and their validity are
discussed.

Consider an axially symmetrical body placed in the test section of
a supersonic tunnel. The test section, instead of providing uniform
flow, may be characterized by a two—dimensional pressure field. In
general, such a pressure field will consist of a longitudinal as well
as a transverse pressure gradient, producing a stream-angle variation
along the tunnel axis.

The difference in symmetry between the body and the field engen—
ders considerable difficulty in the analytic treatment of the problem,
involving a Fourier expansion of the disturbance potential of the body.
Therefore, a relatively simple, linear gradient is assumed: On the axis,
the horizontal component of velocity may equal the original velocity of
the uniform stream. U0, but the transverse gradient may produce a verti-
cal velocity component yielding the desired stream angle variation.