Approximations to the Indicial Response

The exact results given in Eqs. 8.118-8.121 are only valid for short values of time but they are useful because they provide guidance in developing approximations for the indicial functions for all values of time that are also in a more convenient analytic form to derive recurrence solutions to the Duhamel integral or for state-space realizations (see previously). In the general case, the indicial normal force and 1/4-chord pitching moment response to a step change in angle of attack, a, and a step change in nondimensional pitch
rate about the 1 /4-chord, q (= ас/ V), can be represented by the equations

M) + M), (8.122)

M) + у #(*. M) (0.25 – xac), (8.123)

^c(s, M) + j4><(s, M), (8.124)

(8’125)

where фпс refers to the assumed noncirculatory part of the indicial response and фс refers to the assumed circulatory part. While it must be remembered that in the subsonic case this decomposition of the total loading is only an idealization, it is convenient for engineering purposes to handle the problem this way. The (5 term is the Glauert compressibility factor for linearized subsonic flow (i. e., /6 = y/ — M2). In the preceding equations, the initial values of the indicial response functions are given directly by linear piston theory as

It should be noted that in practical applications the linearized value of the lift-curve-slope, 2n/f, can be replaced by the measured value for a given airfoil at the appropriate Reynolds number and Mach number,[34] which is generally denoted by СПа (M). Additionally, the second term in Eq. 8.123 represents the contribution to the pitching moment resulting from a Mach number dependent offset of the aerodynamic center from the 1 /4-chord axis of the airfoil, an effect previously described for the incompressible flow case. As shown in Section 7.7.1, the values of the aerodynamic center, xac, for a given airfoil can be obtained from static airfoil measurements at the appropriate Mach number. Also notice that the second term of Eq. 8.125 represents the induced camber pitching moment resulting from the pitch rate motion, as given by the quasi-steady thin-airfoil result, as also previously described. Therefore, it will be appreciated that the various indicial response functions, denoted by ф, represent the time-dependent behavior of the lift and pitching moment between the initial “piston theory” values at s = 0 and the normal quasi-steady values at s = oo.