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Center for Nonlinear Studies |
Steady fluid solutions can play a special role in characterizing the dynamics of a flow: stable states might be realized in practice, while unstable ones may act as attractors in the unsteady evolution. Unfortunately, determining stability is often a process substantially more laborious than computing steady flows; this is highlighted by the fact that, for several comparatively simple flows, stability properties have been the subject of protracted disagreement (see e.g. Dritschel et al. 2005, and references therein). In this talk, we build on some ideas of Lord Kelvin, who, over a century ago, proposed an energy-based stability argument for steady flows. In essence, Kelvin’s approach involves using the second variation of the energy to establish bounds on the growth of a perturbation. However, for numerically obtained fluid equilibria, computing the second variation of the energy explicitly is often not feasible. Whether Kelvin’s ideas could be implemented for general flows has been debated extensively (Saffman & Szeto, 1980; Dritschel, 1985; Saffman, 1992; Dritschel, 1995). We recently developed a stability approach, for families of steady flows, which constitutes a rigorous implementation of Kelvin’s argument. We build on ideas from bifurcation theory, and link turning points in a velocity-impulse diagram to exchanges of stability. We further introduce concepts from imperfection theory into these problems, enabling us to reveal hidden solution branches. Our approach detects exchanges of stability directly from families of steady flows, without resorting to more involved stability calculations. We consider several examples involving fundamental vortex and wave flows. For all flows studied, we obtain stability results in agreement with linear analysis, while additionally discovering new steady solutions, which exhibit lower symmetry.
Host: Balu Nadiga, CCS-2, balu@beasley.lanl.gov