Predicting vortex-ring boundaries using dynamic inflow methods

Perhaps the most accurate predictions of the flight conditions that lead to vortex-ring can be obtained by considering so-called dynamic inflow [2.22]. Dynamic inflow attempts to account for the effect of the vortex-ring condition on the inflow character­istics through the whole rotor. Wolkovitch [2.23] analysed the vortex-ring state by considering a slipstream that is surrounded by a protective tube of vorticity, see Fig. 2.18.

It can be seen that the vortices descend at an average speed of 0.5 v — g or vi/2— Vg sin у relative to the rotor, whereas the rotor itself descends at a speed Vg sin у. Thus for low rates of descent the vortex-ring will move away from the rotor and steady flow can exist. If, however, the glide slope is too steep the relative velocity of the vortices will fall to zero and they will remain within the rotor disk resulting in the vortex-ring condition. This critical condition can be expressed in terms of a critical glide speed ^гіt:

Kdt 2 sin у

or normalizing using the induced velocity in the hover (vih): Vcrit sin у Vv

v

or:

(2.19)

To express Equation (2.19) in a more convenient form it is necessary to rewrite it in terms of horizontal velocity (Vf = Vg cosy) and vertical velocity (Vv = Vg siny). This is achieved by use of momentum theory, since:

T = 2p AvV’

Since noting that thrust and weight must balance both in a steady descent and in the hover (neglecting the effects of download and vertical drag), we can write:

vV ‘=^-7 = v2 2pA h

Now from Fig. 2.19 it can be seen that for a gliding rotor: V ‘ = W2 + (Vv – v, )2

Thus:

v2h = vi Vf + (Vv – v,)2 v2h = v, Vv2 + VV – 2Vvv, + v2 v4h = v2 (vf2 + v2 – 2vvv, + v2)

Now from Equation (2.18) the vortex-ring condition is entered when: 2Vv = v,. Hence the lower vortex-ring boundary will be represented by:

v4h = 4 v2 (vf2 + vv2)

1 = 4^2^2 + 4^4

Thus:

22^4 + 22^2^2 – 1 = 0

If the rate of descent is allowed to increase and the blade pitch is reduced the helicopter will eventually enter the windmill-brake state. Wolkovitch [2.23] recommended that the upper vortex-ring boundary be represented by:

1.4v

q = —

Again we can recast this equation, since:
< = v2(Vf + V – 2Vvvi + v2)

or:

1 = v2(^2 + q2 – 2^v + v2)

Therefore substituting for q using Equation (2.21) and simplifying:

302q4 + 702q2^2 – 492 = 0 (2.22)