In Section 5.2, flutter analysis was conducted using an aerodynamic theory for steady flow. The lift and pitching moment used were functions only of the instantaneous
pitch angle, в. On deeper investigation, however, it is easy to see that the angle of attack is not simply equal to в. For example, recalling that the airfoil reference point is plunging with velocity h, at least for small angles, we can justify modifying the angle of attack to include the effect of plunge; viz.

a = в + U (5.81)

where this follows from an argument similar to the one used in Section 4.2.5 regarding the influence of aircraft roll on the wing’s angle of attack. However, we must be cautious about such ad hoc reasoning because there may be other effects of the same order that we are overlooking.

Indeed, there are other effects of equal importance that must be included. Fung (1955) suggested an easy experiment to demonstrate that things are not so simple as indicated by Eq. (5.81): Attempt to rapidly move a stick in a straight line through water and notice the results. In the wake of the stick, there is a vortex pattern, with vortices being shed alternately from each side of the stick. This shedding of vortices induces a periodic force perpendicular to the stick’s line of motion, causing the stick to tend to wobble back and forth in your hand. A similar phenomenon happens with the motion of a lifting surface through a fluid and must be accounted for in unsteady-aerodynamic theories.

We can observe that lift and pitching moment consist of two parts from two physically different phenomena: noncirculatory and circulatory effects. Circulatory effects are generally more important for aircraft wings. Indeed, in steady flight, it is the circulatory lift that keeps the aircraft aloft. Vortices are an integral part of the process of generation of circulatory lift. Basically, there is a difference in the velocities on the upper and lower surfaces of an airfoil. Such a velocity profile can be represented as a constant velocity flow plus a vortex. In a dynamic situation, the strength of the vortex (i. e., the circulation) is changing with time, as are both the magnitude and direction of the relative wind vector because of airfoil motion. However, the circulatory forces of steady-flow theories do not include the effects of the vortices shed into the wake. Restricting our discussion to two dimensions and potential flow, we recall an implication of the Helmholtz theorem: The total vorticity will always vanish within any closed curve surrounding a particular set of fluid particles. Thus, if a clockwise vorticity develops about the airfoil, a counterclockwise vortex of the same strength must be shed into the flow. As it moves along, this shed vortex changes the flow field by inducing an unsteady flow back onto the airfoil. This behavior is a function of the strength of the shed vortex and its distance away from the airfoil. Thus, accounting for the effect of shed vorticity is, in general, a complex undertaking and would necessitate knowledge of each vortex shed in the flow. However, if we assume that the vortices shed in the flow move with the flow, then we can estimate the effect of these vortices.

Noncirculatory effects, also called apparent mass and inertia effects, are sec­ondary in importance. They are generated when the wing has nonzero acceleration
so that it must carry with it some of the surrounding air. That air has finite mass, which leads to inertial forces opposing its acceleration.

In summary, then, unsteady-aerodynamic theories need to account for at least three separate physical phenomena, as follows:

1. Because of the airfoil’s unsteady motion relative to the air, the relative wind vector is not fixed in space. This is only partly addressed by corrections such as in Eq. (5.81). The changing direction of the relative wind changes the effective angle of attack and thus changes the lift.

2. As Fung’s experiment shows, the airfoil motion disturbs the flow and causes a vortex to be shed at the trailing edge. The downwash from this vortex, in turn, changes the flow that impinges on the airfoil. This unsteady downwash changes the effective angle of attack and thus changes the lift.

3. The motion of the airfoil accelerates air particles near the airfoil surface, thus creating the need to account for the resulting inertial forces (although this “apparent-inertia” effect is less significant than that of the shed vorticity). The apparent-inertia effect does not change the angle of attack but it does, in general, affect both lift and pitching moment.

Additional phenomena that may affect flutter but which are beyond the scope of this text include three-dimensional effects, compressibility, airfoil thickness, flow separation, and stall.

In this section, we present two types of unsteady-aerodynamic theories, both of which are based on potential-flow theory and take into account the effects of shed vorticity, the motion of the airfoil relative to the air, and the apparent-mass effects. The simpler theory is appropriate for classical flutter analysis as well as for the к and p-к methods. The other is a finite-state theory cast in the time domain, appropriate for the eigenvalue analysis involved in the p method as well as for the time-domain analysis required in control design.