Bifurcation sequence of two-dimensional Taylor-Green vortex via vortex interactions: Evolution of energy spectrum

The vorticity dynamics of the two-dimensional (2D) Taylor-Green vortex (TGV) problem is investigated in its multi-cellular configuration by solving the incompressible Navier-Stokes equation for long time intervals using a pseudo-spectral method. This helps follow the vorticity dynamics of periodic free shear layer flows by solving an extremely accurate algorithm to explain vortex interactions that lead to vortex stripping (forward cascade), merger, and reconnection (inverse cascade) during various stages of evolution of periodic arrangements of a large number of TGV vortical cells. This latter aspect has been adopted so as not to be affected by the periodicity constraints of a single periodic cell and the various imposed symmetries that attenuate disturbance growth. The analytic solution of the TGV provides the initial condition and the spatially accurate Fourier spectral method enables one to track the first instability of the initial doubly periodic vortices. Despite a plethora of studies following the primary instability to relate it with transition to turbulence and the subsequent decay of turbulence in the literature, the topic of bifurcation sequence for periodic TGV is rare, and that is one of the main aims of the present research. Instead of restricting one's attention on a single periodic TGV cell, here it is purposely reported for multiple cells of the TGV in both directions, without invoking any asymmetries extraneously. For such an ensemble, one can study various vortical interactions giving rise to atypical energy spectra, a topic that has also been seldom addressed to distinguish between successive instabilities that can upon a conjecture, lead to transition and subsequent relaminarization, versus the bifurcation sequences leading from one equilibrium state to subsequent ones. The present study shows the dominance of the latter for 2D TGV at post-critical Reynolds number.