Computational Aerodynamics of Ordered Vortex Flows by H.W.M. Hoeijmakers

By H.W.M. Hoeijmakers

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3 5 ) , the v e l o c i t y induced by t h e l i n e v o r t e x first. -38- and, in the narrow-core quasi-cylindrical approximation, for the strength of the vortex r(x) - r(x ) + (x-x )r'(x ) + ... o o o one obtains: -1 " " * x « ( X ) - T 1 / T(X) *• dx v ° 4r \r\3 x-0 where 1 and I.

16a) -21- where C is the_2D counterpart of S and s m tends to infinity. In Eq. 16a) R - Ï -3j(t), the contour element dC is defined in Eq. 10e) and t is a parameter along C . From Eq. 16a) it is concluded that in case the integrated source strength equals zero the perturbation" velocity potential remains bounded at infinity. The velocity potential due to a doublet distribution in 2D follows in a similar manner from Eq. 16b) i s defined as given in Eqs. ( 3 . l i d and e) and shown in F i g . 3 .

The filament replaces the inner portion of the spiral. This leaves us with the finite-length vortex sheet, a feeding sheet of constant doublet strength (in the cross-flow plane s ■ constant) and a disconti­ nuity in the doublet strength at the position of the vortex filament. For a double-branched rolled-up vortex core a similar model applies, see Fig. 14. Replacing the inner part, with the high curvatures and gradients of the doublet distribution, by the vortex filament gives rise to two feeding sheets, each connecting the vortex to one edge of the cut vortex sheet.

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