Generalized Kutta Condition

There can be no pressure difference across the wake. So, for x>b, Cp(x,t)=0. i.e.,
The first term on the rhs describes the bound-circulation contribution, and the second term describes the wake portion.
Multiply by  ; use  . thus, the generalized Kutta condition can be written as:

 

 

Expressions in terms of Laplace Transforms

To deal with cases of general motions, it is useful to express Cp etc. in a tranformed domain rather than in the time domain. This is done through the Laplace transform, which is a linear operator. 
. Thus transforming the Kutta condition gives
Take t=0 to be at an instant before the flow disturbances start, hence the initial condition terms are zeroes.
, and  .

 

Special Case of Simple Harmonic Motion

If all disturbances are simple harmonic motion (i.e., one frequency), all flow parameters must vary with time in a simple harmonic manner as well, since the governing equations are linear. Thus for example,
, so that  . In this case, 
In the tranformed domain, the expression for Cp is:

, or,  where  etc.

Substituting for  (this takes a few pages of derivation, see Bisplinghoff et al.), the final expression is:

 

where  and

is the Theodorsen function. The functions H0 and H1 are Hankel functions.

Note:

The limit as s tends to zero corresponds to the steady-state limit (for periodic, this means  ).
 

. Thus the first term in  is the "quasi-steady" pressure: corresponds to the effect of the bound vorticity.

where the transformed normal-wash in the surface boundary condition is:

. As u tends to zero,  . Also, 

Thus the second term in Cp corresponds to the reaction of the fluid particles adjacent to the surface being accelerated. This is called the "apparent mass" term. Since it can be finite when u=0, it is a non-circulatory component.

The non-circulatory pressure distribution can be modeled using sources and sinks. (see Bisplinghoff et al.)

The third term is due to wake vorticity.

Domains of unsteady aerodynamics for conventional aircraft at subsonic speeds:

1) Quasi-steady: bound vorticity only. This is typically adequate for full-scale aircraft oscillations below 2Hz. Here the reduced frequency is much less than 1. This is what is done in conventional dynamic stability analyses.

2) Quasi-unsteady aerodynamics: roughly 2 to 10Hz. Here the reduced frequency is of the same order as 1. The wake-induced effects must be included, but it is not necessary to include the apparent mass effect.

3) Full unsteady aerodynamics (potential flow). Here all 3 terms are needed, since the reduced frequency is substantially greater than 1.For full-scale aircraft, this regime involves fluctuations faster than about 10 Hz.
 

Lift and Moment Coefficients in the Laplace Transformed Domain

Quasi-steady Coefficients

As seen above, the apparent mass term can be computed independent of the wake, etc. The wake-induced effects can be included using the Theodorsen function.Thus, if the quasi-steady coefficients for lift and moment can be calculated for a given problem, the rest of the problem can be solved easily.
and  . Compare with the thin airfoil theory in Anderson, "Fundamentals of Aeronautics", or in Bertin & Smith, "Aerodynamics for Engineers"
 
 

Given the above, the full lift and moment coefficients (excluding the steady part!) can be expressed as:

and 

Thus the Theodorsen function is seen to be the ratio of Unsteady Circulatory Lift to Quasi-Steady Lift, and is less than or equal to 1.

, where  ; 
 

Quasi-steady coefficients for simple motions

Uniform Normalwash Distribution
. Substituting,  ; 

Linear Normalwash Distribution

; 

Steady Angle of Attack

Camber line equation is:
Normal wash amplitude is:  (constant).
, so that  ;
;  . Location of the center of pressure is:

i.e., center of pressure is at quarter-chord, as we know.
 
 

Simple Harmonic Vertical Translation

Camber line is now  where  is a constant.
Normalwash amplitude is: Hence,  and  ;
and  . 

 

Analogy with Quasi-Steady Aerodynamics:

Effective Angle of Attack can be derived as  . This has only an imaginary part, showing that the lift is 90 degrees out of phase with the displacement. An analogy with the quasi-steady case can thus be used to obtain lift and moment derivatives for thin airfoils in simple harmonic vertical tranalation, by simulating a thin airfoil at a steady scaled angle of attack. This is useful in experimental programs.
 

Steady Symmetric Parabolic Camber

The equation to the mean camber line is:  . Normalwash amplitude is:  is a linear distribution.  ;  ;  ;  ;  . Center of pressure for this camber distribution is at midchord.
Simple Harmonic Pitch

Camber line equation is:  , and the normalwash amplitude becomes:

.

Note: Normalwash consists of a linear variation due to pitch rate, and a constant part due to the instantaneous pitch angle.

Linear part:  . Coefficients become:  ;  ;  ;  . The c.p. is at midchord, similar to the steady parabolic camber case. Thus,

Effective Camber ae*due to pitch rate can be found from:  , so that