Vortex Instability and Transient Growth

The dynamics of vortex flow is studied theoretically and numerically. Starting from a local analysis, where the perturbation in the vortex flow is Fourier decomposed in both radial and azimuthal directions, a modified Chebyshev polynomial method is used to discretize the linearized governing operator. The spectrum of the operator is divided into three groups: discrete spectrum, free-stream spectrum and potential spectrum. The first can be unstable while the latter two are always stable but highly non-normal. The non-normality of the spectra is quantitatively investigated by calculating the transient growth via singular value decomposition of the operator. It is observed that there is significant transient energy growth induced by the nonnormality of continuous spectra. The non-normality study is then extended to a global analysis, in which the perturbation is decomposed in the radial or azimuthal direction. The governing equations are discretized through a spectral/hp element method and the maximum energy growth is calculated via an Arnoldi method. In the azimuthally-decomposed case, the development of the optimal perturbation drives the vortex to vibrate while in the stream-wise-decomposed case, the transient effects induce a string of bubbles along the axis of the vortex. A further transient growth study is conducted in the context of a co-rotating vortex pair. It is noted that the development of optimal perturbations accelerates the vortex merging process. Finally, the transient growth study is extended to a sensitivity analysis of the vortex flow to inflow perturbations. An augmented Lagrangian function is built to optimize the inflow perturbations which maximize the energy inside the domain over a fixed time interval.