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National Advisory Committee for Aeronautics, Report - Three Dimensional Transonic Flow Theory Applied to Slender Wings and Bodies

The present paper re-examines the derivation of the integral
equations for transonic flow around slender wings and bodies of
revolution, giving special attention to conditions resulting from
the presence of shock waves and to the reduction of the relations to
the special forms necessary for the discussion of sonieflow, that
is, flow at free-stream Mach number 1. In the vicinity of the
body, the disturbance field is then shown to consist of a two-
dimensional disturbance field extending laterally and a longi-
tudinal field that depends on the streamwise growth of cross-
section area. This result extends Oswatitsch’s equivalence rule
to lifting cases, provided the angle of attach: is small relative to
the thickness ratio.

The correctness of the analysis is checked
by examination of Yoshihara’s numerical solution for sonic
flow around a slender, circular cone-cylinder and this solution
is checked, in turn, by comparison with experimental results of
Solomon. An example is presented in which the general result
is applied to calculate pressure and integrated forces on a family
of slender, elliptic cone-cylinders. An expression is derived
which permits the ready calculation of the difierence in drag of
two slender bodies having the same longitudinal distribution of
cross-section area. Classes of wings and bodies are described
for which the difi’erence in drag is zero and the Whitcomb area
rule applies. Experimental data for such a family of wings of
rectangular plan form are examined and it is shown that theory
and experiment are in good accord.

The equations governing transonic flows are known and
well established by favorable comparisons with experiment
(see ref. 1 for a resumé). The difficulties arising as a result
of the nonlinearity and mixed character of the differential
equation for the potential, however, have prevented the
rapid advancement of the analysis such as has occurred in
recent years with both subsonic and supersonic theory.

This is particularly true for three-dimensional transonic
flows and, as a result, perhaps greater than usual effort has
gone into the determination and utilization of relations
between solutions. The first of these to be advanced was
the transonic similarity rule which pertains to the pressures
and forces on aflinely related wings (refs. 2, 3, and 4) and
bodies of revolution (ref. 5). A second relation is the area
rule established empirically by Whitcomb (ref. 6) which
states that “near the speed of sound, the zero-lift drag rise
of thin low-aspect—ratio wing-body combinations is primarily
dependent on the axial distribution of cross-sectional area
normal to the air stream.”