Towards a separation of the elements in turbulence via the analyses within MPDFT

The PDFs for energy dissipation rates created in a high resolution from $4096^3$ DNS for fully developed turbulence are analyzed in a high precision with the PDF derived within the formula of multifractal probability density function theory (MPDFT). MPDFT is a statistical mechanical ensemble theory constructed in order to analyze intermittent phenomena through the experimental PDFs with fat-tail. By making use of the obtained w-PDFs created from the whole of the DNS region, analyzed for the first time are the two partial PDFs, i.e., the max-PDF and the min-PDF which are, respectively, taken out from the partial DNS regions of the size $512^3$ with maximum and minimum enstropy. The main information for the partial PDFs are the following. One can find a w-PDF whose tail part can adjust the slope of the tail-part of a max-PDF with appropriate magnification factor. The value of the point at which the w-PDF multiplied by the magnification factor starts to overlap the tail part of the max-PDF coincides with the value of the connection point for the theoretical w-PDF. The center part of the min-PDFs can be adjusted quite accurately by the scaled w-PDFs with a common scale factor.

💡 Research Summary

The paper presents a high‑precision statistical analysis of energy‑dissipation‑rate probability density functions (PDFs) obtained from a state‑of‑the‑art direct numerical simulation (DNS) of fully developed turbulence on a 4096³ grid. The authors employ the multifractal probability density function theory (MPDFT), a statistical‑mechanical ensemble framework they previously developed, to model the intermittent, fat‑tailed PDFs that characterize turbulent dissipation.

MPDFT separates the turbulent field into a coherent component (the singular, scale‑invariant structures that generate the fat tail) and an incoherent component (the viscous dissipation that breaks scale invariance). Accordingly, each PDF is decomposed into a tail part, dominated by the coherent contribution, and a central part, where both coherent and incoherent contributions coexist. The tail is derived analytically from a maximum‑entropy distribution of the singularity exponent α, using either Rényi or HCT entropy, leading to a PDF of the form P⁽ⁿ⁾(α) ∝