Article: Luzzatto-Fegiz P and Williamson CHK (2012). “Structure and stability of the finite-area von Karman street.” Physics of Fluids 24(6).
Abstract: By using a recently developed numerical method, we explore in detail the possible inviscid equilibrium flows for a Karman street comprising uniform, large-area vortices. In order to determine stability, we make use of an energy-based stability argument (originally proposed by Lord Kelvin), whose previous implementation had been unsuccessful in determining stability for the Karman street [P. G. Saffman and J. C. Schatzman, “Stability of a vortex street of finite vortices,” J. Fluid Mech. 117, 171-186 (1982)]. We discuss in detail the issues affecting this interpretation of Kelvin’s ideas, and show that this energy-based argument cannot detect subharmonic instabilities. To find superharmonic instabilities, we employ a recently introduced approach, which constitutes a reliable implementation of Kelvin’s stability ideas [P. Luzzatto-Fegiz and C. H. K. Williamson, “Stability of conservative flows and new steady fluid solutions from bifurcation diagrams exploiting a variational argument,” Phys. Rev. Lett. 104, 044504 (2010)]. For periodic flows, this leads us to organize solutions into families with fixed impulse I, and to construct diagrams involving the flow energy E and horizontal spacing (i.e., wavelength) L. Families of large-I vortex streets exhibit a turning point in L, and terminate with “cat’s eyes” vortices (as also suggested by previous investigators). However, for low-I streets, the solution families display a multitude of turning points (leading to multiple possible streets, for given L), and terminate with teardrop-shaped vortices. This is radically different from previous suggestions in the literature. These two qualitatively different limiting states are connected by a special street, whereby vortices from opposite rows touch, such that each vortex boundary exhibits three corners. Furthermore, by following the family of I = 0 streets to small L, we gain access to a large, hitherto unexplored flow regime, involving streets with L significantly smaller than previously believed possible. To elucidate in detail the possible solution regimes, we introduce a map of spacing L, versus impulse I, which we construct by numerically computing a large number of steady vortex configurations. For each constant-impulse family of steady vortices, our stability approach also reveals a single superharmonic bifurcation, leading to new families of vortex streets, which exhibit lower symmetry.
(C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4724307]