Abstract

Problems involving the evolution of coherent fluid structures arise within a wide range of situations, including planetary flows (Carton, 2001), fluid turbulence (Dritschel et al., 2008), aquatic animal propulsion (Dabiri, 2009), and wind turbine wakes (Sørensen 2011). Steady solutions can play a special role in characterizing the dynamics: stable flows might be realized in practice, while unstable ones may act as attractors in the unsteady evolution of the flow.

In this talk, we consider the problem of finding steady states of the two-dimensional Euler equation from topology-preserving rearrangements of a given vorticity distribution. We begin by briefly reviewing a range of available numerical methodologies. We then focus on a recently introduced technique, which enables the computation of steady vortices with (1) compact vorticity support, (2) prescribed topology, (3) multiple scales, (4) arbitrary stability, and (5) arbitrary symmetry. We illustrate this methodology by computing several families of vortex equilibria. To the best of our knowledge, the present work is the first to resolve nonsingular, asymmetric steady vortices in an unbounded flow. In addition, we discover that, as a limiting solution is approached, each equilibrium family traces a clockwise spiral in a velocity-impulse diagram; each turn of this spiral is also associated with a loss of stability. Such spiral structure is suggested to be a universal feature of steady, uniform-vorticity Euler flows. Finally, we examine the problem of selecting vorticity distributions that accurately model practically important flows, and build a constructive procedure to compute attractors of the Navier-Stokes equations. We consider an example involving a vortex pair with distributed vorticity, and obtain good agreement with data from Direct Numerical Simulations.