Horseshoe Vortex

A simplified case of the vortex ring is the horseshoe vortex. In this case the vortex line is assumed to be placed in the x-y plane as shown in Fig. 10.26. The two trailing vortex segments are placed parallel to the x axis at у = ya and at У ~Уъ> and the leading segment is placed parallel to the у axis between the points {xa, уa) and (xa, yb). The induced velocity in the x-y plane will have only a component in the negative z direction and can be computed by using Eq. (2.69) for a straight vortex segment:

-Г

w(x, y, 0) = — (cos /3, – cos /32) 4 Ли

where the angles and their cosines are shown in the Fig. 10.26. For example, for the semi-infinite filament shown,

For the vortex segment parallel to the x axis, and beginning at у=уь, the corresponding angles are

For the finite-length segment parallel to the у-axis,

FIGURE 10.26

Nomenclature used for deriving the influence of a horseshoe vortex element.

where the finite-length segment does not induce downwash on itself.

The velocity potential of the horseshoe vortex may be obtained by reducing the results of a constant strength doublet panel (Section 10.4.2) or by integrating the potential of a point doublet element. The potential of such a point doublet placed at (jc0, y0, 0) and pointing in the z direction, as derived in Section 3.5 (or in Eq. (10.110)) is

4л r3

where r = V(jc – xQ)2 + (y — y0)2 + z2. To obtain the potential due to the horseshoe element at an arbitrary point P, this point doublet must be integrated over the area enclosed by the horseshoe element:

<b = ZT p Г________________ zdxо_________

4/r і Уо1 [(x – x0f + (y – y0)2 + г2Га The result is given by Moran,51 p. 445, as

Note that we have used Eq. (B.4a) from Appendix В to evaluate the limits of the first term.