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National Advisory Committee for Aeronautics, Technical Notes - Application of Two Dimensional Vortex Theory to the Prediction of Flow Fields Behind Wings of Wing Body Combinations at Subsonic and Supersonic Speeds

A theoretical investigation has been made of a general method for
predicting the flow field behind the wings of plane and cruciform wing
and body combinations at transonic or supersonic speeds and slender con-
figurations at subsonic speeds. The wing trailing-vortex wake is repre—
sented initially by line vortices distributed to approximate the spanwise
distribution of circulation along the trailing edge of the exposed wing
panels. The afterbody is represented by corresponding image vortices
within the body. Two-dimensional line—vortex theory is then used to
compute the induced velocities at each vortex and the resulting displace-
ment of each vortex is determined by means of a numerical stepwise inte-
gration procedure. The method was applied to the calculation of the
position of the vortex wake and the estimation of downwash at chosen tail
locations behind triangular-wing and cylindrical—body combinations at
supersonic speeds. The effects of such geometric parameters as aspect
ratio, angle of attack and incidence, ratio of body radius to wing semi-
span, and angle of bank on the vortex wake behind wings of wing-body
combinations were studied. The relative importance of wing vortices,
the corresponding image vortices within the body, and body crossflow in
determining the total downwash was assessed at a possible tail location.

It was found that the line-vortex method of this report permitted
the calculation of vortex paths behind wings of winngody combinations
with reasonable facility and accuracy. A calculated sample wake shape
agreed qualitatively with one observed experimentally, and sample results
of the line-vortex method compared well with an available exact crossflow—
plane solution. An empirical formula was derived to estimate the number
of vortices required per wing panel for a satisfactory computation of
downwash at tail locations. It was found that the shape of the vortex
wake and the ultimate number of rolled-up vortices behind a wing depend
on the circulation distribution along the wing trailing edge. For the
low—aspect-ratio plane wing and body combinations Considered, it appeared
that downwash at horizontal tail locations is largely determined except
near the tail4body juncture by the wing vortices alone for small ratios
of body radius to wing semispan, and by the body upwaSh alone for large
values of that ratio.