Minimum wave drag for arbitrary arrangements of wings and bodies Page: 3 of 12
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NACA TN 3530
drag. Problems of this type have been considered by E. W. Graham and
his colleagues who give, for example, the optimum distribution of lift
within a spherical region.
If the region R is restricted to the plan form S of a planar wing,
then problems of a type previously discussed by the present writer are
obtained (refs. 5 and 6). In connection with the latter problems it was
found that all distributions of lift or volume satisfying the given
requirements could be characterized by relatively simple conditions. The
present paper describes briefly the extension of these conditions to three-
dimensional regions and the additional conditions required.
As is usual in linearized-flow problems it will be assumed that the
disturbance field of the airfoils and bodies can be produced by the action
of a distribution of sources and "lifting elements" or horseshoe vortices.
One of the difficulties associated with these problems is the determina-
tion of the actual geometric shapes produced by the distribution of singu-
larities. In the present analysis the relation between the body shapes
and the singularities is not known nor determined in detail. For slender
bodies or thin airfoils closed within the region R it-can be assumed
that the total volume is proportional to the first moment of the source
distribution with respect to a plane perpendicular to the flight direc-
tion, whereas the total lift is proportional to the total strength of
the lifting elements.
Suppose a region R together with a distribution of singularities
such as source's or lifting vortices is given. (See fig. 1.) Then by
Hayes' theorem (ref. 7), the drag will be unchanged by a reversal of the
whole system. (The geometry of the flow, including that of the airfoils
and bodies, will be changed by the reversal but the total lift and the
total volume will not.) Then the drag may be computed by means of a fic-
titious "combined disturbance field" obtained by superimposing the dis-
turbances in the forward and the reversed motion. The perturbation veloc-
ities in this combined field may be denoted by
2ii = uf + ur
2 = vf + yr
2~ = wf +r
An arrangement of sources or lifting elements or their combination which
yields the minimum drag is then characterized by the conditions2
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Jones, Robert T. Minimum wave drag for arbitrary arrangements of wings and bodies, report, February 1956; (https://digital.library.unt.edu/ark:/67531/metadc60742/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.