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Why don't we worry about the airfoil wake in steady potential flow? The answer is that there is no net vorticity in such a wake, as shown in the figure below. There is a region of momentum deficit formed as the boundary layers from the top and bottom surfaces merge, and if you go inside this region you will certainly experience shear and hence vorticity. However, looking from outside this region, we see that there is no net vorticity in the wake, since the sense of rotation of the upper and lower parts of the wake are opposite.

In the case of the airfoil suddenly changing circulation, shown below the steady-flow figure, the wake indeed has a net vorticity, which makes it roll up into "starting vortex". In this case, we can imagine that the bottom part of the wake has a higher velocity than the top, so that it curls around.

What happens when there is a sudden change in the angle of attack of the airfoil?

The downstream stagnation point stays at the trailing edge (see Note 1 below). Thus, the bound circulation

changes from

. Net vorticity is now seen across the wake (see Note 2 below). The wake rolls up into a "starting vortex" of strength

, which convects at

Note 1: Here we assume that there is no time lag in the response of

to angle of attack a. This breaks down when we try to explain dynamic lift phenomena at high angles of attack and very high rates of pitch in incompressible flow. It is also inadequate in compressible flow, where the freestream velocity is of the same order of magnitude as the speed of sound.

Note 2: Where does the vorticity come from?

From dissipation of flow kinetic energy in the boundary layer, due to viscous effects, and the resulting loss in momentum. It is associated with a drop in stagnation pressure in the boundary layer, and an increase in entropy.

Unsteady Effects on Lift Per Unit Span

The steady-state lift per unit span is:

The quasi-steady lift change due to change in angle of attack (or, change in bound circulation) is:

. Note that from now on, when we speak of "lift", we mean "change in lift due to change in..". There is always a steady-state lift term to be remembered when the final answer is calculated.

As seen from Kelvin's theorem, when the bound vorticity changes, and equal and opposite amount of vorticitymust be shed into the wake. The shed vorticity in the wake induces velocity at the airfoil (as can be calculated using the Biot-Savart Law), opposing the effect of the quasi-steady lift change. This "lift deficiency" effect decays as the shed vortex moves downstream. This is seen in the figure below.

A convenient description of this wake-induced lift loss has been developed by Theodorsen(1935) for simple harmonic motion of a thin airfoil in a uniform flow. The Reduced Frequency of the motion is represented as:

. The wake-induced lift loss can be represented as a complex function of k, C(k)= F(k) +iG(k)

Reference: Theodorsen, Theodore, "General Theory of Aerodynamic Instability and the Mechanism of Flutter". N.A.C.A. Technical Report No. 496, Washington, DC, 1935.

Apparent Mass Effect

When you push against air, you have to pay for the acceleration of the air. The mass of air being accelerated by motion of the airfoil, perpendicular to its chord, is estimated as the mass of air within the circular cylinder whose diameter is the chord. Thus,

This is called "non-circulatory lift", since it has nothing to do with vorticity. Also, note that the freesream velocity does not appear in the above expression: The non-circulatory lift term can exist even when the freestream velocity is zero! This is important when considering the ability of insects to hover. Let's consider the phase of this lift with respect to the motion of the airfoil.

Consider simple harmonic vertical motion of an airfoil.

describes the displacement z(t).

is the velocity. This is 90 degrees out of phase with the displacement.

is the acceleration. This is 90 degrees out of phase with velocity, and 180 degrees out of phase with displacement. Thus the non-circulatory lift is 180 degrees out of phase with the displacement. These can be expressed also as follows:

Thus, total lift per unit span is given by:

for simple harmonic motion.

or,

. The quantity within parentheses is called the "circulatory lift" term (note that the steady-state lift is also circulatory, but here the discussion is about the components of the time-varying lift.

Thus, the time-varying "circulatory lift" is given as:

for small-amplitude, simple harmonic motion of a thin airfoil in steady flow. (Remember that we should add the steady lift to get the total lift)

(quasi-steady limit),

. This function C(k) is tabulated, and is called the Theodorsen Lift-Deficiency Function.

For small k, an approximation of the tabulated function is:

where

. This is "Euler's constant". See the linked graph for a comparison of the approximate expression with the tabulated function.

Results for other kinds of motion can be derived conveniently from this basis. For example,

(a) continuous motions can be synthesized as Fourier series of simple harmonic motions at different frequencies.

(b) the response to an impulsive airfoil motion in steady flow can be calculated using the Wagner function, whose calculation includes the Theodorsen function.

(c) the Kussner function is used to calculate airfoil response in an unsteady freestream (note: this is not the same as unsteady airfoil motion in steady flow).

In the following, we include a perspective on the derivation of the time-varying lift, using the Theodorsen function and the apparent mass. This touches upon some of the essential features of airfoil aerodynamics, but does not go into the detailed, explicit derivation of the Theodorsen function

Small-Amplitude Oscillations of a Thin Airfoil in a Steady Flow

Reference: Katz and Plotkin, p. 453 -

Assume that the (x,z) frame of reference moves to the left at a constant speed

. The time-dependent position of the chord is represented by vertical displacement h(t), positive in the +z direction, and by the time-varying pitch angle

, pitching about an axis through the chord at

. The chordline is then represented by

. The frame (x,z) does not rotate. Small-amplitude motion is introduced through

, where

;

. Then

;

The downwash (by the surface boundary condition, see Note A) is:

where

is the wake effect. This can also be written as:

From thin airfoil theory, we can substitute for

and get coefficients An: (see Note A2)

The airfoil surface boundary condition is

where B(x,y,z,t)=0 describes the surface. If B(x,y,z,t) is expressed as

, we get

;

, etc.

NOTE A2: Carleman-Schwarz Inversion

In the following, the nomenclature used is from the old books on unsteady aerodynamics. The quantity b is semi-chord, equivalent to c/2. The coordinate reference x = 0 is at midchord. The leading edge is at x = -b, and the trailing edge at +b. The normal velocity at the surface comes from 3 sources: