Statistics and dynamics of vortices in turbulence

Now we are interested in investigating

1. identification, visualization and understanding of swirling vortices in homogeneous turbulence
2. time development of skeletons of vortices and comparison with the time evolution of enstrophy density
3. statistics of the physical characteristics of vortex cores

by applying the SPMS method to analyze numerical results of a turbulence simulation. In this article, as the first step of our research, we observe some typical aspects of the vortex skeletons and cores throughout a decaying simulation of homogeneous and isotropic turbulence. The initial condition of these simulation was given by a fixed energy spectrum

The phases of Fourier components of the velocity of the fluid are given by random numbers. Number of the grid points is and the viscosity is set as . A decaying simulation is executed by solving the Navier-Stokes equation with the triple-periodic boundary condition by using the spectral and Runge-Kutta-Gill schemes.

In Fig. , we show skeletons obtained by the SPMS method and isosurfaces of the enstrophy density at t = 144. The flow field has been fully developed by this time. Skeletons are observed at the outside of the isosurfaces as well as at the inside of the isosurfaces. It suggests that tubular structures of the enstrophy density represents swirling motion (as many people expect) because skeletons are observed just at the center of these isosurfaces, and that there are many swirling motions which are not identified by the isosurfaces of the enstrophy density. The threshold of isosurfaces was determined so that the of the entire volume is contained inside the isosurfaces. Note that the criterion of the -volume is quite empirical and that we do not have any objective criterion to determine the threshold. The lack of objectivity to give an appropriate threshold is a serious difficulty which is common with many methods of vortex definition.[7, 8] It is obvious that existence of swirling vortices around skeletons are objective. Furthermore, the SPMS method identifies individual vortices and visualize each of them by a skeleton. It implies that we can investigate nature of vortices one by one while such a way of investigation is quite difficult by any methods of visualization adopting isosurfaces as the enstrophy density, Q-, - and -definitions. ( The names Q, and represent scalars which are used to define vortex cores. See Jeong and Hussain[7] to find explanation about these methods. )

In Fig. , typical vortex cores identified by the SPMS method at t = 144 are shown. About of the volume and about of the enstrophy density are contained inside vortex cores at this time. We have found that cross-sections of these vortex core with long skeletons tend to be larger than those of shorter ones. If we assume that the cross-sections are approximated by circles, a typical diameter of the cross-sections is given by about eight times of the Kolmogorov length scale. Note that the simulation box is full-filed if we display all of vortex cores simultaneously, to fail to understand individual vortex structures. It is a problem common with many identification methods. Although Q-, - and -definitions give objective thresholds to define vortex cores without any ambiguity, isosurfaces of these quantities cover too much regions and fail to extract coherent structures. However, the SPMS method can extract individual vortex cores as shown in Fig.. Thus we can understand various property of vortex structures by this method.

Finally, we note that skeletons investigated here can be traced smoothly in the time direction if we observe them in the animation. Visualization of smooth motions of skeletons (and therefore vortex cores) may provide us a possibility to understand dynamics of turbulence as an ensemble of dynamical motions of vortices. Temporal-evolution of individual vortices (a set of skeleton and vortex core) may be traced in order to study dynamics of vortex such as elongation of skeletons or interaction with other vortices.