Effect of Unsteady Forcing on the Sinusoidal Instability of Vortex Wakes

An analytical model of the vortex wake instability subject to forcing by unsteady lift fluctuations has been developed. These lift fluctuations can be caused either by variations in angle of attack or by flying through atmospheric turbulence. The effect of turbulence in the fluid itself, already studied by Crow and Bate, has been reviewed and incorporated into the analysis. A simple procedure is provided to estimate the time at which contact occurs between the two wake vortices as a function of certain characteristics of the forcing inputs. The behavior of the instability is somewhat altered by these forcing terms. In particular, it was found that contact between vortices can occur at much shorter wavelengths than predicted for the unforced case. Furthermore, in the presence of forcing, the minimum completion time for the instability does not correspond to the wavenumber for which the amplification rate is a maximum because the length of the vortex trajectories is also an important factor. Finally, the stability boundary is extended to larger wavenumbers by the presence of forcing, and the boundary location depends on the forcing level. factors are unsteady wing loading and the presence of disturbances in the flowfield. Each of these effects has already received some attention, as described below. The effect of an unsteady load distribution was investigated by Bilanin and Widnall7 for the case of constant net lift. The effect of differentially operated flaps was studied theoretically and experimentally and was shown to be a possible means of hastening the completion of the sinusoidal instability. Chevalier8 showed in flight tests that periodic angle-of-attack variations forced the instability and reduced the time required for the vortices to come into contact. Ap- parently the important case of varying the net lift has not yet been studied theoretically. In practice, load variations could arise either by the lifting surface encountering disturbances or by continual corrective motions applied by the pilot. The effect of turbulence in the fluid in which the sinusoidally deforming wake is embedded was investigated by Crow and Bate.9 The cases of very strong and very weak turbulence levels were considered and the results were used to form a composite curve for the wake lifespan (defined statistically) as a function of the turbulent energy dissipation rate. The turbulent eddies were assumed to lie in the Kolmogorov inertial subrange. The results show that the presence of turbulence will reduce the mean lifespan of the vortex wake, however the effect is relatively weak for low turbulence levels. Portions of the present work reflect the influence of Ref. 9. The present study has a somewhat different emphasis than the earlier basic work on the sinusoidal instability,1"3'6 which had as a major concern the identification of the underlying mechanism and the prediction of the amplification rate and stability boundaries. The present work builds on the previous results and seeks to describe the factors which force or provide initial conditions for the instability. Characterization of these factors allows the prediction of a definite time for the instability to go to completion. The result is therefore more precise than simply saying that the instability goes to com- pletion in a time on the order of the time scale associated with the amplification rate. Furthermore, the relative importance of the various factors which force the instability can be compared. This approach has also been taken in Refs. 7 and 9.