Modal and non-modal stability of two-dimensional Taylor-Green vortices

The linear stability of the two-dimensional Taylor-Green vortices at moderate Reynolds number Re = 2500 is studied by modal and non-modal stability analysis. The properties of modal stability depend on the symmetry type of the disturbances. The stability of penetrating disturbances depends weakly on the axial wavenumber, while non-penetrating disturbances are subjected to the elliptic instability which depends crucially on the wavenumber. The features of transient growth are revealed by non-modal stability analysis. Its mechanism for small optimization time is the stretching near the hyperbolic stagnation points; it does not depend strongly on the symmetry type and the wavenumber. For large optimization time, however, the transient growth is controlled by the strain field around the elliptic stagnation points as the amplified state is similar to unstable modes of elliptic instability. The optimization time at which the properties of the transient growth change is smaller than that of a vortex pair studied by Donnadieu et al (2009 Phys. Fluids 21 094102).