The interaction of two identical circular viscous vortex rings starting in a side-by-side configuration is investigated by solving the Navier–Stokes equation using a spectral method with 643 grid points. This study covers initial Reynolds numbers (ratio of circulation to viscosity) up to 1153. The vortices undergo two successive reconnections, fusion and fission, as has been visualized experimentally, but the simulation shows topological details not observed in experiments. The shapes of the evolving vortex rings are different for different initial conditions, but the mechanism of the reconnection is explained by bridging (Melander & Hussain 1988) except that the bridges are created on the front of the dipole close to the position of the maximum strain rate. Spatial structures of various field quantities are compared. It is found that domains of high energy dissipation and high enstrophy production overlap, and that they are highly localized in space compared with the regions of concentrated vorticity. The kinetic energy decays according to the same power laws as found in fully developed turbulence, consistent with concentrated regions of energy dissipation. The main vortex cores survive for a relatively long time. On the other hand, the helicity density which is higher in roots of bridges and threads (or legs) changes rapidly in time. The high-helicity-density and high-energy-dissipation regions overlap significantly although their peaks do not always do so. Thus a long-lived structure may carry high-vorticity rather than necessarily high-helicity density. It is shown that the time evolution of concentration of a passive scalar is quite different from that of the vorticity field, confirming our longstanding warning against relying too heavily on flow visualization in laboratory experiments for studying vortex dynamics and coherent structures.
The direct algebraic method for constructing travelling wave solutions of nonlinear evolution and wave equations has been generalized and systematized. The class of solitary wave solutions is extended to analytic (rather than rational) functions of the real exponential solutions of the linearized equation. Expanding the solution in an infinite series in these real exponentials, an exact solution of the nonlinear PDE is obtained, whenever the series can be summed. Methods for solving the nonlinear recursion relation for the coefficients of the series and for summing the series in closed form are discussed. The algorithm is now suited to solving nonlinear equations by any symbolic manipulation program. This direct method is illustrated by constructing exact solutions of a generalized K~V equation, the Kuramoto-Sivashinski equation and a generalized Fisher equation.
Kolmogorov's theory for turbulence, proposed in 1941, is based on a hypothesis that small-scale statistics are uniquely determined by the kinematic viscosity and the mean rate of energy dissipation. Landau remarked that the local rate of energy dissipation should fluctuate in space over large scales and hence should affect small-scale statistics. Experimentally, we confirm the significance of this large-scale fluctuation, which is comparable to the mean rate of energy dissipation at the typical scale for energy-containing eddies. The significance is independent of the Reynolds number and the configuration for turbulence production. With an increase of scale r above the scale of largest energy-containing eddies, the fluctuation comes to have the scaling r −1/2 and becomes close to Gaussian. We also confirm that the large-scale fluctuation affects small-scale statistics.
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