Dynamic Inflow

The mabi term, therefore, is meant to represent the additional reaction force on the rotor disk because of the accelerating inflow. However this concept assumes an equivalence between the force on a solid disk accelerating in a stagnant fluid and the force on a fluid accelerating through a permeable actuator disk. Such an equivalence is certainly not rigorous but the derivation can proceed on this basis. Using the blade element theory (see Section 3.3), the elemental thrust including blade flapping is given by

dT = ^NbpcC[a (б – Q2r2dr. (10.110)

Assuming a rectangular blade planform (i. e., c(r) = c) and uniform inflow (i. e., u,(r) = uf) the total rotor thrust can be found by integrating the above equation to get

т = [ат = ъИьрсС^кг{в-т-ъ)-

Equating the two thrust expressions (Eq. 10.108 and 10.111), an equation governing the dynamic inflow through the rotor can be derived. Notice that the inflow dynamics is coupled with the blade flapping dynamics, and a second equation governing the blade flapping angle can be found by moment equilibrium about the flapping hinge, as described previously in Chapter 4. In nondimensional form Eq. 10.111 becomes

★

P

3

2

h + i

в

3

Ct

(10.112)

Therefore, the governing equation for the inflow dynamics can be written as

A,

2

(10.113)

2A{ А,- + – /3

To determine the approximate time constant of the developing flow, Eq. 10.ІІ5 can be linearized about a mean operating state using А,- = A; + <$A; and Ст = Ст + 8Ct, which after simplification and neglecting terms of О (Ski)2 gives

Oaii+ai=(i)sCr – (ioil6)

This is an ODE that relates the change in inflow, 5A;, to the change in rotor thrust, 8Ct – The time constant of this dynamic system is

0.849

4AIT2’

(10.117)