Improved predictions
The ‘Wolkovitch’ boundaries although based on a detailed representation of the vortexring condition fail to match empirical data. Experience gained from wind tunnel experiments and flight test suggests that the vortexring condition can be escaped by allowing a forward velocity component to develop. Thus the lower boundary should show that as the horizontal velocity (q) is increased higher rates of descent (q) are required to enter the vortexring state. This can be achieved if the Wolkovitch approach is modified to account for the effect of forward speed on the wake geometry, see Fig. 2.20.






































Therefore the lower vortexring boundary is given by:
2
ц = v——
VJ
2 1 4
p = 7 – V6
The upper vortex boundary is obtained in an analogous manner, by replacing v with kv in Equation (2.23). As mentioned above, Wolkovitch recommended k = 1.4 as denoting the edge of the fully developed vortex ring condition. Peters and Chen [2.22], on the other hand, prefer k= 2 as indicating the condition beyond which no vortex is present anywhere in the streamtube. Hence Equation (2.23) becomes:
31
p2 = 2 ^2v) — ц2 — ^(2v)2 = 3^ — ц2 — 2v2
Once again from momentum theory, 1 = v2(p2 __ ц2 — 2^ + v2), thus: 1 = v2(3^ — ц 2 + ц2 — 2v2 — 2цv + v2)
1 = v2^v — v2) = — v4
1
Ц = v + —
v3
Consequently:
Therefore the upper vortexring boundary is given by:
1
Ц = v + —
v3
These boundaries are shown in Fig. 2.21. Clearly the lower boundary approximates to the empirical trend whereby with forward speed higher rates of descent are required to enter the vortexring state. There are, however, two issues that remain unresolved. First the upper and lower boundaries should predict the same forward velocity, above which vortexring can be completely avoided. Secondly the upper boundary is based on entry to the windmillbrake state (Vv = 2vih in the hover) whereas the lower indicates entry into fully developed vortexring (V = 0.7vih in the hover). In order to resolve these problems the lower boundary must be further modified to start at p = 0, ц = 0. This is achieved by noting that a vortex will appear in the streamtube when (ba)/ Ib< 0. Thus the lower boundary should be given by:
p2 + ц(ц — v)
V^2 + (v — ц)2
Fig. 2.21 Vortexring boundaries using modified Wolkovitch analysis. 
Applying momentum theory gives:
1 = v2(^v — Ц + Ц — 2qv + v2)
1 = v2(v2 — цу)
1
Ц = v ——
Consequently:
^2=3v (v—УЗ)—(v—УЗ)— 2у2=У2—У6
Therefore the complete criteria, see Fig. 2.21, for the vortexring state are:
1
Ц = v±—
2 1 1
^ = 7 — У6
These criteria can be used to predict a complete and coherent boundary for the onset of the vortex ring condition [2.22].
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