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.

No abstract

The deviations from the Kolmogorov 1941 laws of inertial range of turbulence are investigated using the results from the direct numerical simulations of an unforced flow starting from a high-symmetry initial condition by Kida [J. Phys. Soc. Jpn. 54, 2132 (1985)]. The resolution is 3003 points (12003 with symmetries, maximum wavenumber 400 after dealiasing), and the Taylor scale Reynolds number is in the order of 100. The scaling exponents of the pth order longitudinal and lateral structure function (for p between 2 and 16) are computed using different methods with particular focus on a recent method by Benzi and collaborators [Phys. Rev. E 48, R29 (1993); Europhys. Lett. 32, 709 (1995)]. Both longitudinal and lateral scaling exponents deviate considerably from Kolmogorov 1941 (K-41) scaling laws, the lateral deviating much more than the longitudinal. A systematic methodology (strain–enstrophy state) is developed to relate the K-41 deviations to different structures in the field. Enstrophy-dominated structures are found to contribute mainly to the deviations in lateral direction whereas the strain-dominated structures to longitudinal direction, albeit in an imbalanced proportion, the lateral deviations being much stronger. Structures whose enstrophy and strain are comparable in magnitude contribute to deviations in both directions. Results are compared to several intermittency models and experiments. Special focus is given to the recent She–Lévêque model [Phys. Rev. Lett. 72, 336 (1994)] whose predictions gave very good agreement if compared to the longitudinal exponents. The model is rewritten for a family of free parameters, giving predictions as good as the original one. The lateral scaling exponents disagree with both the She–Lévêque model and the experimental results (from longitudinal velocity measurements) suggesting that the dominant contribution to intermittency can only be detected from the lateral structure function measurements.

The three-dimensional time evolution of two orthogonally offset cylindrical vortices of equal strength is simulated by solving the hyperviscosity-regularized incompressible Navier-Stokes equations. A Fourier pseudospectral method with a time-split integration scheme is used for the solution. Four runs with different Reynolds numbers ranging between 690-2100 are performed, each with a resolution of 963 collocation points. The sequence of important physical processes and the evolution of local and global quantities such as vorticity, velocity, and mean-square strain rate are presented. It is found that the growth rate of the maximum vorticity is at most exponential. The Reynolds number dependence of the time scale of reconnection, the vorticity growth rate, and the time at which the maximum vorticity is attained are examined and differences between the present results and Saffman's essentially two-dimensional model predictions are encountered and elucidated. The distributions of the eigenvalues cx, /3, y and the corresponding eigenvectors s,, sP , sy of the rate of strain tensor S, are calculated at different times. It is found that as the mean-square strain rate E increases during the evolution, sg and the vorticity vector o are perfectly aligned and ,8> 0 in high E regions. Strong temporal, spatial, and Reynolds number dependence of the strain fields is also seen. Evidence is presented that, during reconnection, the vorticity growth in newly forming bridges takes place in the vicinity of the upper stagnation line segment of the vortex dipole due to the nature of the vortex stretching term. Also examined is the initial finger formation and it is found that the initial nonuniform axial stretching and the displacement of the vortex cores due to a lift force play an important role in this process.

The problem of whether a spontaneous singularity can occur in finite time in an incompressible inviscid fluid flow is addressed. As suggested by previous numerical simulations, candidate flows are restricted to be invariant under the octahedral group of symmetries and to have a compact vortex tube in the fundamental domain. It is shown that in such a flow the image vorticity contributes strongly to the axial strain rate on the fundamental in a way which is only weakly proportional to the curvature of the vortex lines. Analysis of a model flow shows that axial strain rate scales as the inverse square of the distance to the origin, and that the velocity field forms a topological trap in which the vortex tube is accelerated towards the origin – a degenerate critical point. Evidence from simulations supports these findings. These features suggest that linear strain rate/vorticity coupling can occur in a finite-time pointwise collapse of such symmetric flows.