Study of two elements constituting turbulence by a multifractal theory for probability density functions-Through the analyses of 4096³ DNS

The probability density functions (PDFs) of energy dissipation rates created from the whole of the DNS region of the size 4096³ (named whl-PDFs) for various sizes of coarse graining areas are analyzed by multifractal PDF theory (MPDFT). Furthermore, other PDFs named max-PDF and min-PDF for various sizes of coarse graining areas, created respectively from the partial DNS regions of the size 512³ with maximum and minimum enstrophy, are also analyzed by making use of the whl-PDF. It is shown that there is a whl-PDF with an appropriate size of coarse graining area, whose slope of the tail part can adjust the slope of the tail-part of a max-PDF. On the other hand, there is a whl-PDF whose center part can adjust quite accurately the center part of a min-PDF. Note that the partial DNS region with maximum (minimum) enstrophy is the region of maximally dense (rare) vortexes, and therefore contributes mainly to the tail (center) part of the whl-PDF. It is found that these adjustments are systematic with respect to the sizes of coarse graining areas.