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National Advisory Committee for Aeronautics, Technical Notes - Approximate Theory for Calculation of Lift of Bodies, Afterbodies, and Combinations of Bodies

An expression is developed for the lift of a slender afterbody in
terms of the "slender-body" approximate potential at the after end in
conjunction with a suitably calculated value of the potential at the
forward end. The failure of the usual "slender-body" theory to predict
any lift on a slender cylindrical afterbody is thereby corrected. The
same expression is used to compute the part of the interference lift
generated.by the presence of a neighboring body, due to the interference
upwash. Another expression is developed to compute the remainder of
the interference lift, due to an interference pressure gradient. The
lift is determined for a cone-cylinder body, a cylindrical afterbody
of a slender wing~body combination, and three combinations of bodies
to illustrate the method.

In the search for better aerodynamic configurations for supersonic
flight, the missile shape has become more complex. One of the problems
arising is the effect of interference between bodies which occurs, for
example, when the propulsive unit is mounted external to the fuselage.
An adaptation of slender-body theory for obtaining a simple approxima—
tion of the interference lift between such bodies was developed at
the NASA Lewis laboratory and is presented in this report. This approxi—
mate theory is also applied for the calculation of the lift of an afterbody,
which is herein defined as the portion of the body behind the wing trailing
edge for a winngody combination or as the portion of the body aft of the
nose section for a wingless body.

Slender—body theory as originally developed by Munk in studying
the lift of airships (reference 1) has proved useful in predicting the
lift of low-aspect-ratio wings, slender bodies, and slender wing-body
combinations at supersonic speeds (references 2, 5, and 4). However,
slender—body theory yields the unrealistic result that a cylindrical
afterbody of a wing-body combination and a cylindrical afterbody of
a body carry no lift.

Although slender—body theory may be grossly in error on the lift
of a slender afterbody, the theory closely approximates the correct
value of the part of the surface potential at the rear of the afterbody
proportional to the angle of attack. An expression utilizing this fact
is developed herein for the afterbody lift in terms of the slender-body
potential at the after end in conjunction with a suitably calculated
value of the potential at the forward end. The more accurate methods of
calculation of greater labor (for example, linearized theory) are thereby
limited to the forward or nonslender portion of the body.