| Most theoretical and computational studies of turbulence in Navier-Stokes fluids and/or guiding-center plasmas have been carried out in the presence of spatially periodic boundary conditions. In view of the frequently-reproduced result that two-dimensional and/or MHD decaying turbulence leads to structures comparable in length scale to a box dimension, it is natural to ask if periodic boundary conditions are an adequate representation of any physical situation. Here, we study, computationally, the decay of two-dimensional turbulence in a Navier-Stokes fluid or guiding-center plasma in the presence of circular rigid walls with either no-slip or stress-free boundary conditions. The method is wholly spectral, and relies on a Galerkin approximation by a set of functions which obey two boundary conditions at the wall radius (closely related to "Stokes eigenfunctions" of the slow flow equation, and analogues of the Chandrasekhar-Reid functions). It is possible to explore Reynolds numbers up to the order of 1250, based on an rms velocity and a box radius. It is found that decaying turbulence is altered significantly by the rigid boundaries. First, strong boundary layers serve as sources of vorticity and enstrophy and enhance the early-time energy decay rate, for a given Reynolds number, well above the periodic boundary condition values. More importantly, in the no-slip case angular momentum turns out to be an even more slowly decaying ideal invariant than energy, and to a considerable extent governs the dynamics of the decay. Angular momentum must be taken into account, for example, in order to achieve quantitative agreement with the prediction of maximum entropy, or "most probable," states. These are predictions of conditions that are established after several eddy turnover times but before the energy has decayed away. Angular momentum will cascade to lower azimuthal mode numbers, even if absent there initially, and the angular momentum modal spectrum is eventually dominated by the lowest mode available. When no initial angular momentum is present, no behavior that suggests the likelihood of inverse cascades is observed. In the stress-free case, angular momentum is a rigorous constant of the motion and we are always at liberty to work in a coordinate system in which it vanishes. The topology of the late-time flow towards which the stress-free cases evolve is totally different from that achieved in the no-slip case with finite angular momentum. |
