Theodorsen’s Unsteady Thin-Airfoil Theory

Theodorsen (1934) derived a theory of unsteady aerodynamics for a thin (meaning a flat-plate) airfoil undergoing small, simple harmonic oscillations in incompressible flow. The derivation is based on linear potential-flow theory and is presented in detail along with mathematical subtleties in the textbook by Bisplinghoff, Ashley, and Halfman (1955). The lift contains both circulatory and noncirculatory terms, whereas the pitching moment about the quarter-chord is entirely noncirculatory. According to Theodorsen’s theory, the lift and pitching moment are given by

where the generalized forces are given in Eqs. (5.24). The function C(k) is a complex­valued function of the reduced frequency k, given by

where H, P(k) are Hankel functions of the second kind, which can be expressed in terms of Bessel functions of the first and second kind, respectively, as

tff>(k) = Jn(k) – iYn(k)

The function C(k) = F(k) + iG(k) is called Theodorsen’s function and is plotted in Figs. 5.9 and 5.10. Note that C(k) is real and equal to unity for the steady case

(i. e., к = 0). As к increases, we find that the imaginary part increases in magnitude whereas the real part decreases. As к tends to infinity, C(k) approaches 1/2. However, for practical situations, к rarely exceeds unity. Hence, the plot in Fig. 5.9 only extends to к = 1. The large к behavior is shown in Fig. 5.10. When any harmonic function is multiplied by С(к), its magnitude is reduced and a phase lag is introduced. An example of this phenomenon is given herein.

A few things are noteworthy concerning Eqs. (5.82). First, in Theodorsen’s theory, the lift-curve slope is equal to 2n. Thus, the first of the two terms in the lift is the circulatory lift without the effect of shed vortices multiplied by С(к). The multiplication by С(к) is a consequence of the theory having considered the effect of shed vorticity. The noncirculatory terms (i. e., the second term in the lift as well as the entire pitching-moment expression) depend on the acceleration and angular acceleration of the airfoil and are mostly apparent-mass/apparent-inertia terms. The circulatory lift is the more significant of the two terms in the lift. Note that the coefficient of h in the lift is the mass per unit length of the air contained in an infinitely long circular cylinder of radius b. This quantity reflects how much air is imparted an acceleration by motion of the airfoil.

For steady flow, the circulatory lift is linear in the angle of attack; however, for unsteady flow, there is no single angle of attack because the flow direction varies along the chordline as the result of the induced flow varying along the chord. However, just so we can discuss the concept for unsteady flow, it is helpful to introduce an effective angle of attack. For simple harmonic motion, it can be inferred from Theodorsen’s theory that an effective angle of attack is

As shown in Section 5.5.2 by comparison with the finite-state aerodynamic model introduced therein, a is the angle of attack measured at the three-quarter chord based on an averaged value of the induced flow over the chord. Recall that in the case of steady-flow aerodynamics of two-dimensional wings, the angle of attack is the pitch angle в. Here, however, a depends on в as well as on h, в, and к. Because of these additional terms and because of the behavior of С(к), we expect changes in magnitude and phase between в and a. These carry over into changes in the magnitude and phase of the lift relative to that of в. Indeed, the function С(к) is sometimes called the lift-deficiency function because it reduces the magnitude of the unsteady lift relative to the steady lift. It also introduces an important phase shift between the peak values of pitching oscillations and corresponding oscillations in lift.

When we see the dots over h and в in the lift and pitching-moment expressions, it is tempting to think of them as time-domain equations. However, the presence of С(к) is nonsensical in a time-domain equation. Therefore, Theodorsen’s theory with the С(к) present must be recognized as valid only for simple harmonic motion.

Note that an approximation of Theodorsen’s theory in which С(к) is set equal to unity is called a “quasi-steady” thin-airfoil theory. Such an approximation has value only for cases in which к is restricted to be very small. For slow harmonic oscillations
or slowly varying motion that is not harmonic, the quasi-steady theory may be used in the time domain.

As an example to show the decrease in magnitude and change of phase, consider that the dominant term in the lift is proportional to a. In the time domain, lift is real and so are a and в. However, when we regard в as harmonic; viz.

в = в exp(iot) (5.86)

then we must realize that to recover the time-domain behavior, we need

в = Щв exp(irnt)] (5.87)

Similarly, we must recover the time-domain behavior of a using the relationship

a = Щ[Є(к)в exp(irnt)] (5.88)

Now, assuming в = 1 so that in the time domain в = cos(ot), we find that
a =Щ[С(к)в exp(iot)]

= Щ[С(к) exp(i ot)]

= F(к) cos(ot) – G(k) sin(ot) (5.89)

= [F2(k) + G2(k)]2 cos(ot – ф)

= |C(k)| cos(ot – ф)

where

tan(<^ = – Gk (5.90)

Because |C(k)| < 1 and ф(к) > 0, having the amplitude of в equal to unity implies that a has an amplitude less than unity; having the peak of в at t = 0 implies a has its peak shifted to t = ф/o. For example, when k = 1/3, C(k) = 0.649739 – 0.174712 i so that |C(k)| = 0.672819, implying a magnitude reduction of nearly 33%, and ф = 15.0506 degrees.

Theodorsen’s theory may be used in classical flutter analysis. There, the reduced frequency of flutter is not known a priori. We can find k at the flutter condition using the method described in Section 5.3. Theodorsen’s theory also may be used in the k and p-k methods, as described in Sections 5.4.1 and 5.4.2, respectively.