Indicial Lift from Pitch Rate

The indicial lift response to a step change in pitch rate q about the 1/4-chord can also be written as the sum of a noncirculatory part, C"^ and a circulatory part, , that is

C„,(s) = C^(s, M) + Ccat(s, M) (8.137)

or in terms of the indicial functions

Numerous references have shown that, for incompressible flow, the chordwise pressure variation on the airfoil is the same as the thin-airfoil loading but is independent of the mode of motion. Therefore, an angular velocity about some point can be considered equivalent to an angular velocity about some other point plus an angle of attack. In particular, the indicial lift for a pitch rate about the 3/4-chord position in incompressible flow is an impulse at s = 0

onH ic 7prn tbf*TV*Q-ftpr $2nVic**mu»nt1 v it finllnwc that frvr inr>nrrmrpccibl<a flnw = rhc

Шіиш V/1UVI.1; ^4^ , 111V/I1VITU X* w »» —- ysq.

In linearized subsonic flow, the circulatory lift lag still remains an intrinsic function of the fluid itself. The chordwise pressure variation remains the same as the steady thin-airfoil loading and is unaffected by the mode of forcing or pitch axis location. Therefore, on a thin airfoil in subsonic flow, the lift always acts at the 1 /4-chord point. In view of the foregoing, it is valid to assume 4>ca{s, M) — <pcq{s, M) for subsonic compressible flow as well as incompressible flow without any loss of rigor. The circulatory part of the lift from pitch rate about the 1 /4-chord, therefore, can be written as

The noncirculatory function can be assumed to be of the form

= = p^). (8.140)

Using Eqs. 8.139 and 8.140 and following the same procedure as before where the slopes of the approximating indicial function and the exact result (in this case, Eq. 8.120) are matched at j = 0 gives the time constant as

T, = = [(1 – М) + 2жрмАф1 + ЛЛ>Ґ “ = K*T‘- <8Л41>