Pressure Variation

The pressure variation through the rotor flow field in the hover state can be found from the application of Bernoulli’s equation along a streamline above and below the rotor disk. Remember that there is a pressure jump across the disk as a result of energy addition by the rotor, so that Bernoulli’s equation cannot be applied between points in the flow across the disk. But the pressure jump is uniform over the rotor disk so the equation can be applied to all streamlines contained within the control volume. For incompressible flow, the Bernoulli equation is an alternative to the energy equation (one of the two is redundant). Applying Bernoulli’s equation up to the disk between stations 0 and 1 produces

PO = Poo = Pi + ^pvf.

Below the disk, between stations 2 and oo, the application of Bernoulli’s equation gives

1 2 1 2 P2 + 2PVi = Poo + £Pw

Because the jump in pressure Ap is assumed to be uniform across the disk, this pressure jump must be equal to the disk loading, T/А, that is,

T

Ap = P2~Pi = — • (2.20)

A

Therefore, we can write

T f 1 2 1 Л / 1 Д 1 2

д = P2 – P – I Poo + – pw – – pvt I – I Poo – – pvt 1 = – pw, (2.21)

from which it is seen that the rotor disk loading is equal to the dynamic pressure in the vena contracta. One ean also determine the pressure just above the disk and just below the disk in terms of the disk loading. Just above the disk the use of Bernoulli’s equation gives

and just below the disk we get

1 2 1 /w2 3 ST

P2 = PO + ^pw – – p{–) = P0 + ? (-) . (2.23)

Therefore, the static pressure is reduced by (T/A) above the rotor disk and increased by |(Г/A) below the disk.