Indicial Lift from Angle of Attack

The lift or normal force response to a unit step change in angle of attack, a, can be idealized as the sum of a decaying noncirculatory part, C”c, and a growing circulatory part, СсПа, that is

Cn.(s) = C£(s, M) + Ccna{s, M) (8.130)

or in terms of the indicial functions

C„„(s) – ~<f>a(s, M) + у #(*, M). (8.131)

Using reciprocal relations, the noncirculatory component of the loading on the airfoil can be theoretically extracted from the total lift response. This was first done by Mazelsky (1952a). The result can be closely approximated as an exponentially decaying function, and so the noncirculatory part of lift for a unit step change in a may be written as –

C£(S, M) = m = L exp, (8.132)

where T’a = T^(M) > 0 is a Mach number dependent time-constant still to be defined. For the growing (circulatory) part of the total indicial response, Mazelsky (1952a, b) appears to have been one of the first investigators to use an exponential approximation of the form

N N

0cCs, M) = 1 — J2 Ane~bnS-, J2 A“ = 1, bn > 0 (8.133)

n—1 И=1

for compressible flow, where the coefficients An and bn will vary as a function of Mach number. These coefficients were obtained indirectly by Mazelsky by relating back into the time domain from numerical results (transfer functions) obtained for oscillating airfoils in the frequency domain. A similar approach has also been suggested by Dowell (1980) to obtain approximations to the indicial response for incompressible flow using Theodorsen’s exact result and for compressible flows by using transfer functions numerically computed using CFD solutions to the unsteady flow problem. In general, indicial type CFD calculations are rare in the published literature, but direct indicial and vertical sharp-edged gust solutions have been performed by McCroskey & Goorjian (1983) using CFD methods. A series of more elaborate direct indicial calculations have been performed by Parameswaran & Baeder (1997) who have computed indicial angle of attack, pitch rate and sharp-edged gust results using an Euler/Navier-Stokes CFD method. These solutions provide good check cases for unsteady airfoil problems that cannot be simulated by experimentation nor amenable to exact analytical treatment. However, these CFD solutions are only available at a relatively high computational cost, and even then they are still subject to certain approximations and limitations.

While Mazelsky’s exponential approximation in Eq. 8.133 may be acceptable for appli­cations in fixed-wing analyses, it is not entirely convenient for a helicopter rotor analysis. This is because each blade station encounters a different local Mach number as a function of both blade radial location and azimuth angle. Therefore, repeated interpolation of the An and bn coefficients will be required between successive Mach numbers to find the locally effective indicial function. Although relatively simple in concept, there is relatively large computational overhead associated with this process. In addition, it must be recognized that when superposition is applied to find the unsteady lift, each exponential term in the series in Eq. 8.186 contributes an additional state or deficiency function. To this end, Beddoes
(1984) and Leishman (1987a, 1993) have assumed that the circulatory lift can be expressed in terms of a two term growing exponential function, but one that scales directly with Mach number alone. The lift function fca(s, M) takes the general form

<pca(s, M) = 1 – Ae~b’p2s – A2e-b2^s A+ A2 = 1, Ъъ b2 > 0, (8.134)

where /3 = Vl — M2, and the A and b coefficients are fixed and independent of Mach number.[35] The use of the fi2 term in the indicial function reflects the effects of compressibility (Mach number) on the growth of the circulatory part of the lift (through the effects of the shed

lags in the growth of lift at higher Mach numbers. Such a behavior is well known from a theoretical standpoint [see Osborne (1973)] as well as from experimental observations with oscillating airfoils. Furthermore, in a practical sense, not only is this simple compressibility scaling approach a computationally efficient way of accounting for compressibility effects in the wake, but it is also more accurate than repeated linear interpolation of the coefficients between discrete Mach numbers.

The coefficients A, A2, b, and b2 must be assumed as initially unknown, and although they could be derived in a number of ways, they can be reliably determined by relating back from frequency domain results using experimental measurements for oscillating airfoils in subsonic flow. Such measurements are widely available; see, for example, Liiva et al. (1968), Wood (1979), and Davis & Malcolm (1980). It is possible to reduce the number of coefficients implicit in the indicial representation in Eq. 8.131 from five to four by obtaining an expression for the noncirculatory time constant, in terms of the coefficients A, A2,b,

and b2. Differentiating the approximation in Eq. 8.131 (using Eqs. 8.132 and 8.134) and the exact result in Eq. 8.118, and equating the gradients at s = 0 gives the time constant in the s domain as

T'{M) =————————————————- . (8.135)

“ 2(1 — M) + 2nfM2{Ab + A2b2)

Using the results Ta = (c/2V)T^ and M = V/a gives the time constant in the t domain as

Ta = [ (1 – M) + лрмАфі + A2b2)Y C~ = KaTit (8.136)

where 7} = с/a. The advantage of taking this approach is that, regardless of the values of the coefficients Ai, A2t bi, and b2, the noncirculatory constant, Ta, will always be adjusted give the correct initial value and slope of the total indicial response as given by exact linear theory in Eq. 8.118. The continuity between the initial impulsive and succeeding circulatory loading is then obtained using linear superposition, as given by Eq. 8.131.