Secondary instability mechanisms in the wake of a flat plate

We extend the work of Sutherland and Peltier (Phys. Fluids 6 (1994) 1267) and investigate numerically the three-dimensional (3D) secondary stability of a wake forming behind a thin flat plate. The primary flow is a Kármán vortex street numerically computed from the two-dimensional (2D) even instability of a parallel wake based on the Bickley velocity profile. Considering the symmetries of the von Kármán Street, the 3D modes are classified into two families, whether symmetric or antisymmetric. For each family, we determine the leading eigenmodes using a Krylov method. The growth rate curves show that both the most unstable symmetric and antisymmetric modes are stationary and present a maximum of amplification for a wavelength of the order of the primary vortex spacing. The maximum growth rate, corresponding wavelength and cutoff wavelength are well predicted by the elliptic instability of the vortex core. The eigenmode structure of the most unstable wavenumber is centered in the core and is typical of the elliptic instability. The hyperbolic instability of the braid region gives a growth rate five times larger and a cutoff two times higher than the ones computed. As recently discussed for mixing layers by Caulfield and Kerswell (Phys. Fluids 12 (5) (2000) 1032), this is not surprising since the hyperbolic instability applies for an unbounded hyperbolic flow. When the region of hyperbolic flow is bounded, intense transient growth is generated, but when time goes to infinity, the instability becomes small or even dies out. Finally, good qualitative and quantitative agreement is found with the experiments previously done by Julien, Lasheras and Chomaz (J. Fluid Mech. 479 (2003) 155) on the secondary instability in the wake of a flat plate for the symmetry selection, the most amplified wavenumber and growth rate.