Reduced Frequency and Reduced Time

One important parameter used in the description of unsteady aerodynamics and unsteady airfoil behavior is the reduced frequency. This parameter is used to characterize the degree of unsteadiness of the problem. It can be shown that the reduced frequency appears when nondimensionalizing the Navier-Stokes equations. Alternatively, it can be shown using dimensional analysis that the resultant force F on an airfoil of chord c, oscillating at angular frequency со in a flow of velocity V, can be written in functional form as

the proof of which is the basis of Question 8.1. As noted previously in Section 7.3, the resultant force, F, depends on the Reynolds number, Re, and the Mach number, M, but now the reduced frequency, k, of the flow is a third parameter to be considered. The reduced frequency is normally defined in terms of the airfoil semi-chord, b = c/2, so that

cob coc ~V ~ 2V

For к = 0, the flow is steady. For 0 < к < 0.05, the flow can be considered quasi­steady; that is, unsteady effects are generally small, and for some problems they may be neglected completely. Usually flows with characteristic reduced frequencies of 0.05 and above are considered unsteady, and the unsteady terms in the governing equations cannot be routinely neglected. Problems that have characteristic reduced frequencies of 0.2 and above are considered highly unsteady, and the unsteady terms, such as those associated with acceleration effects, will begin to dominate the behavior of the airloads.

For a helicopter rotor in forward flight the reduced frequency at any blade element is an ambiguous parameter because the local sectional velocity (which appears in the denomi­nator of the reduced frequency expression) is constantly changing. However, a first-order approximation for к can give useful information about the degree of unsteadiness found on a rotor and the necessity of modeling unsteady aerodynamic effects in any form of analysis. Consider first the unsteady effects induced by rigid blade flapping, for which it has been shown in Section 4.7 that the first flap frequency is about 1.05 £2 for an articulated rotor. Then the reduced frequency, k75, at the 75% radius location (r = 0.75) will be

assuming for simplicity that the local velocity at the blade element is just the rotational veloc­ity, rQR. For a helicopter rotor with a blade aspect ratio R/c > 10, then k7s > 0.07, which
is in the unsteady range. Also, because the reduced frequencies increase further inboard on the blade owing to the lower values of local sectional velocity, к may become relatively large. Consider further the first elastic torsion mode, which is typically about 3-4£2. In this case, at the tip the reduced frequency associated with airloads generated by torsional dis­placements is in excess of 0.2. At these reduced frequencies, there is a significant amplitude and phasing introduced into the airloads by the effects of the unsteady aerodynamics, and the modeling of unsteady aerodynamics is critical if erroneous predictions of the airloads are to be avoided.

It should be appreciated that the previous reduced frequency calculations are only very approximate and serve only to illustrate the potential significance of unsteady effects and the need to model such effects in predicting the airloads in rotor problems. For quantification of more transient problems, the concept of a single reduced frequency in terms of characterizing the degree of unsteadiness of the problem begins to lose its significance. Under these circumstances it is normal to use reduced time, s, where

(8.4)

which represents the relative distance traveled by the airfoil through the flow in terms of airfoil semi-chords during a time interval t. It has been found useful to characterize many of the events occurring in unsteady aerodynamics, such as dynamic stall or blade encounters with blade tip vortices, in terms of a reduced time parameter.