Decaying two-dimensional turbulence in square containers with no-slip or stress-free boundaries

We report results of direct numerical simulations of decaying two-dimension al (2D) turbulence inside a square container with rigid boundaries. It is s hown that the type of boundary condition (no-slip or stress-free) determine s the flow evolution essentially. During the initial (0 less than or equal to t less than or equal to 0.2 root Re) and intermediate (0.2 root Re less than or equal to t less than or equal to 3 root Re) stages of decaying 2D t urbulence (t congruent to 1 is comparable with an eddy turnover time, Re is the Reynolds number of the flow), the decay scenario for simulations with no-slip boundary conditions can be understood from turbulent spectral trans fer and selective decay. A third mechanism can be recognized for t greater than or equal to 3 root Re: A decay stage where diffusion dominates over no nlinear advection, i.e., spectral transfer is then absent in favor of self- similar decay. The present results show that at presently accessible Reynol ds numbers and computation times, laboratory experiments cannot be accurate ly compared with quasi-stationary states from ideal maximum-entropy theorie s or with computed solutions of flows in containers with stress-free bounda ries. The decay which results in rectangular containers with no-slip bounda ries does not yet yield anything that is meaningfully comparable with these formulations. The evolution of the number of vortices V, the average vorte x radius a, the ratio of enstrophy Omega over energy E, and the extremum of vorticity (normalized by root E) have been computed based on ensemble aver aging of the no-slip runs. An algebraic regime has been observed with V(t)s imilar to t(-0.90), a(t) similar to t(0.31), Omega(t)/E(t)similar to t(- 0. 63), and omega(ext)(t)/ root E(t)similar to t(-0 30). Finally, quantities s uch as a measure of the viscous stresses near the boundaries have been comp uted in order to analyze the decay of 2D turbulence in containers with rigi d boundaries. (C) 1999 American Institute of Physics. [S1070-6631(99)01403- 8].