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  to  . 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.

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: 

This is also: 

. 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.

The aspect of interest here is the time-variation of the lift-change term.

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)

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, 

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:

;  ;  ; where 

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.
 

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)

As  (quasi-steady limit),  .

As  ,  . 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 tabulaed function.
 

(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

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  or  , where  ;  . Then  ; 

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

. Obviously,

. Using the transformation from thin airfoil theory,

From thin airfoil theory, we can substitute for  and get coefficients An: (see Note A2)

where  for n=0, and

for n=1,2,3......

Here,  ;  ; and  for n> 2
 

Circulation due to downwash w* is (from thin airfoil theory, see Note A3)

so that  .

Lift per unit span

where  is the steady part.

NOTE A1:

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.

From the definition  we get 

Now, for pitch-plunge motion,

;  . Thus, 

NOTE A2: Carleman-Schwarz Inversion

w_b, the induced velocity of the bound vortex sheet distribution,

w_a, the surface motion, and

l, the wake-induced velocity.

Thus,  . Using the Biot-Savart Law, 

. The integral equation to be solved is thus:  . This can be inverted using the Carleman-Schwarz inversion:

. This satisfies the conditions of  at x =b, and is singular at the leading edge, as required by the inversion formula.

Substitute

. Assume that the wake-induced velocity and the body surface motion can be represented by series:  and  . Substituting,

.

Note:  . Using these,

for n other than zero, and for all n.

.

Substituting, we get

Note:  , so that:

where  for n=0, and  for 

From the above, note that:

NOTE A3:

Circulation due to  :

. The third term is zero, leaving