Unsteady Lifting Force Coefficient

During rapidly changing unsteady motion of an airfoil the aerodynamic response is no longer the timewise slightly changing steady phenomenon.

For example, let us consider a thin airfoil with a chord length of 2b undergoing a vertical simple harmonic motion in a free stream of U with zero angle of attack.

If the amplitude of the vertical motion is h and the angular frequency is ю then the profile location at any time t reads as

Za (t) = heix t (1.5)

If we implement the pure steady aerodynamics approach, because of Eq. 1.3 the sectional lift coefficient will read as zero. Now, we write the time dependent sectional lift coefficient in terms of the reduced frequency k = xb/U and the non­dimensional amplitude h* = h/b.

cl(t) = [-2 ikC(k)h* + k2h* ] л eix г (1.6)

Let us now analyze each term in Eq. 1.6 in terms of the relevant aerodynamics.

(i) Unsteady Aerodynamics: If we consider all the terms in Eq. 1.6 then the analysis is based on unsteady aerodynamics. C(k) in the first term of the expression is a complex function and called the Theodorsen function which is the measure of the phase lag between the motion and aerodynamic response. The second term, on the other hand, is the acceleration term based on the inertia of the air parcel displaced during the motion. It is called the apparent mass term and is significant for the reduced frequency values larger than unity.

(ii) Quasi Unsteady Aerodynamics: If we neglect the apparent mass term in Eq. 1.6 the aerodynamic analysis is then called quasi unsteady aerodynamics. Accordingly, the sectional lift coefficient reads as

cl(t) = [ —2лikC(k)h*] eix‘ (1.7)

Since the magnitude of the Theodorsen function is less than unity for the values of k larger than 0, quasi unsteady lift coefficient is always reduced. The Theodorsen function is given in terms of the Haenkel functions. An approximate expression for small values of k is: C(k) ffi 1 — л k/2+ ik(ln(k/2) + .5772), 0.01 < k < 0.1.

(iii) Quasi Steady Aerodynamics: If we take C(k) = 1, then the analysis becomes a quasi steady aerodynamics to give

cl(t) = [—2 лikh*] eix‘ (1.8)

In this case, there exists a 90o phase difference between the motion and the aerodynamic response.

(iv) Steady Aerodynamics: Since the angle of attack is zero, we get zero lift!

So far, we have seen the unsteady aerodynamics caused by simple harmonic airfoil motion. When the unsteady motion is arbitrary, there are new functions involved to represent the aerodynamic response of the airfoil to unit excitations. These functions are the integral effect of the Theodorsen function represented by infinitely many frequencies. For example, the Wagner function gives the response

to a unit angle of attack change and the Kussner function, on the other hand, provides the aerodynamic response to a unit sharp gust.