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National Advisory Committee for Aeronautics, Report - Determination of Vortex Paths by Series Expansion Technique with Application to Cruciform Wings

A series method of determining two-dimensional vortex paths
is considered and applied to the computation of vortex positions
behind a slender equal-span cruciform wing at any angle of
bank as a function of the distance behind the trailing edge.
Calculated paths are shown for four bank angles. For a
bank angle of 45° comparison is made with the results of a
closed expression given in NACA TN 2605. For other ban]:
angles water-tank experiments provide qualitative comparison.
Satisfactory agreement is found for a sufficient distance downstream
to include most practical missile-tail positions.

The interference forces on an equal-span integrated
cruciform tail behind a slender equal span cruciform wing
are calculated for five angles of bank (including the trivial
case of zero bank) from the cortex positions found by use of
the series.

It is now well established that the vortex wake at the
tail of a slender configuration similar. to those used for
many missiles is often entirely rolled up and that the down-
wash field at the tail can be obtained by use of a single
discrete vortex as an approximation to the vortex wake trailing
from each wing panel. If attention is confined to configura-
tions for which the ideas of conventional slender—body theory
can be used, the problem of determining the steady-state
vortex paths becomes an exact analog of the classical prob—
lem of the motion of a system of parallel rectilinear vortices.

Sacks, in reference 1, has investigated the case of an
equal-span cruciform wing at 45° angle of bank Where the
symmetry of the problem permits the writing of a closed
analytic solution for the vortex paths. The direct extension
of his method to other angles of bank where no such sym—
metry exists does not appear feasible.

In the present paper, in order to avoid the requirement
of symmetry, a series has been developed to define the vor-
tex paths. Paths computed by this method are compared
With the analytic results of Sacks for 45° angle of bank, and
with the results of water-tank experiments for three other
bank angles. Calculations are made of the forces on a tail
due to vortices in the computed positions.