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National Advisory Committee for Aeronautics, Technical Notes - A Linearized Solution for Time Dependent Velocity Potentials Near Three Dimensional Wings at Supersonic Speeds

A source-distribution method is applied to derive a solution for
the time-dependent surface velocity potential of thin finite wings at
supersonic speeds. The solution is illustrated by evaluating the
upwash over the tip of an. arbitrary-plan-boundary wing having a
supersonic and subsonic leading edge. The upwash then gives the
effective sources of the flow field lying between the wing plan
boundary and the foremost Mach waves, which are applied in the
derivation of the wing-surface velocity potential. A simple ample,
the load distribution on a wing whose effective angle of attack is -
changing linearly with time, is included to illustrate the appli-
cations of the derived expressions.

The analysis of the aerodynamic effects in the vicinity of thin
wings at supersonic speeds can be simplified to obtain useful results
by means of the linearized theory. Steady-state or time-independent
solutions have been obtained for enough cases (for ample, refer-
ences l to 27) that the essential features of the load distributions
on various parts of the wing can be determined by graphical or
analytical methods. The time-dependent load distributions are more
difficult to obtain. Such problems include the transient effects
of gusts, changes in angle of attack, skin vibration, and flutter.

A number ‘of investigators have studied two-dimensional time-
dependent flows over thin wings. ‘ These flows are generally included
as special cases of the theory of reference 27. The method of
reference 27 is similar to the steady-state solution of reference 1
and includes three-dimensional or finite wing solutions in cases
where the aerodynamic effects of the bottom and top wing surfaces are
independent. No solutions are known to have been published for
cases involving interaction between the flow over the bottom and
top wing surfaces.