Final states of decaying 2D turbulence in bounded domains: Influence of the geometry

Direct numerical simulations of two-dimensional decaying turbulence in wall bounded domains are presented. The Navier–Stokes equations are solved using a Fourier pseudo-spectral method with volume penalization. Starting from random initial conditions, we study the influence of the geometry of the domain on the flow dynamics, in particular on the long time behaviour. Circular, square, triangular and annular domains are considered and we show how the geometry plays a crucial role regarding the decay scenario towards final states. Three stages can be distinguished: formation of coherent vortices from random initial conditions, vortex wall interactions, and finally relaxation towards a quasi-steady structure. The eigenvalues estimated from the decay rate of both energy and enstrophy depend on the geometry and agree well with the theoretical eigenvalues based on the Stokes mode of the corresponding domain. For the final states we find a linear functional relation between vorticity and streamfunction.