Direct numerical simulation of transition to turbulence from a high-symmetry initial condition

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Abstract
The three-dimensional (3-D) time evolution of a high-symmetry initial condition [J. Phys. Soc. Jpn. 54, 2132 (1985)] is simulated using a Fourier pseudospectral method for Re=1/ν=500, 1000, 2000, and 5000 with an effective resolution of 10243 collocation points (1713 independent modes, maximum wave number kmax=340). It is found that much before the peak enstrophy is reached, there is a short interval when the local quantities increase sharply. It is also found that during this interval, six vortex dipoles (at the origin) and three dipoles (at the π/2 corner) collapse toward two separate vorticity null points at the opposite corners of the domain in a nearly self-similar fashion. The coherent vortices break up afterward, followed by a sharp decrease in local quantities. The singularity analysis shows that, within the limits of the resolution, the maximum vorticity scales approximately as (T−Tc)−1, shortly before the breakup. However, the increase in peak vorticity stops at a certain time, possibly due to viscous dissipation effects. The temporal evolution of the width of the analyticity strip shows that δ approaches zero at a rate faster than exponential, but reaches a minimum value and starts to increase. This suggests that the solution remains uniformly analytic, as is the case in the viscous Burgers equation.

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