Vortex Ring

Based on the subroutine of Eq. (10.116), a variety of elements can be defined. For example, the velocity induced by a rectilinear vortex ring (shown in Fig. 10.24) can be computed by calling this routine four times for the four

(X4<yi. Zi)

FIGURE 10.24

Influence of a rectilinear vortex ring.

segments. Note that this velocity calculation is equivalent to the result for a constant-strength doublet.

To obtain the velocity induced by the four segments of a rectangular vortex ring with circulation Г calculate

(«1, Vi, Wj) = VORTXL (x, y, z, xu yb zu x2, y2, z2, Г)

(u2, v2, w2) = VORTXL (x, y, z, x2, y2, z2, x3, y3, z3, Г)

(u3, v3, w3) = VORTXL (x, y, z, x3, y3, z3, x4, y4, z4, Г)

(u4, v4, w4) = VORTXL {x, y, z, x4, y4, z4, xu yu zu Г)

and the induced velocity at P is

(u, V, w) = (lli * VU Wi) + (u2, v2, w2) + (u3, V3, Wj) + (u4, v4, tv4)

This can be programmed into a subroutine such that

X у z *1 У z2 x2 У2 Z2 *3 Уз z3 X4 y4 z4

In most situations the vortex rings are placed on a patch with i, j indices, as shown in Fig. 10.25. In this situation the input to this subroutine can be abbreviated by identifying each panel by its i, jth corner point.

(u, v, w) = VORING (x, y, z, i, j, Г,,) (10.117a)

From the programming point of view this routine simplifies the scanning of the vortex rings on the patch. However, the inner vortex segments are scanned twice, which makes the computation less efficient. This can be improved for larger codes when computer run time is more important than programming simplicity.

Note that this formulation is valid everywhere (including the center of the element) but is singular on the vortex ring. Such a routine is used in Program No. 12 in Appendix D.

FIGURE 10.25

The method of calculating the in­fluence of a vortex ring by adding the influence of the straight vortex seg­ment elements.