A General Framework for Destabilizing Neutrally-Stable Flows Applied to Aircraft Wake Vortices

Moore & Saffman (1975), Tsai & Widnall (1976), and others considered forced, resonant instabilities of a neutrally-stable aircraft wake vortex. Generally, the wake vortex was modeled as a q-vortex. In the previous studies, the q-vortex was forced by a finite-amplitude perturbation of the form of a linear strain field or as a boundary layer mode. Using resonance theory, the previous authors showed that the finite forcing could perturb the eigenvalues of a degenerate pair of neutrally-stable eigenmodes of the q-vortex such that one had a positive growth rate. Here we introduce a general framework based, in part, on the degenerate perturbation theory used in quantum mechanics, to show how to perturb a q-vortex, or any other neutrally stable flow, such as Couette-Taylor flow, to produce unstable eigenmodes. Using a perturbed eigenmode, along with the perturbed q-vortex as an initial condition in a code that solves Euler’s equation, we show that the amplitudes of the perturbed eigenmode initially grow linearly with the rates predicted by our degenerate perturbation theory. Our initial-value code is then used to determine which perturbed modes saturate with weak amplitudes (i.e., have large Landau coefficients) and which grow large enough to potentially de-stabilize a vortex wake.