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This study introduces a new family of volumetric flatness factors which give a rigorous parametric description of the phenomenon of intermittency in fully developed turbulent flows. These quantities gather information about the most “active" part of a velocity field at each scale ℓ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell $$\end{document}, and allows one to define a dimension function p→Dp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p \rightarrow D_p$$\end{document} that recovers intermittency correction to the structure exponents ζp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\zeta _p$$\end{document} in an explicit way. In particular, the predictions of the Frisch–Parisi multifractal formalism can be recovered in a systematic and rigorous way. Within this framework we identify active regions that carry the most energetic part of a velocity field at a given scale ℓ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell $$\end{document}. A threshold for what constitutes being active is defined explicitly. Active regions have proven to be experimentally observable in our previous joint work (Nguyen et al. in Phys Rev E 99:053114, 2019), and been shown to capture concentration of the energy cascade as ℓ→0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell \rightarrow 0$$\end{document}, in Cheskidov and Shvydkoy (SIAM J Math Anal 46(1):353–374, 2014). We present several examples of fields which exhibit arbitrary multifractal spectrum within theoretically permitted limitations. At the same time we demonstrate, with the use of a probabilistic argument, that a random field is expected to produce the classical K41 spectrum in the limit ℓ→0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ell \rightarrow 0$$\end{document}. Intermittent deviations from K41 theory are estimated at any finite scale also. Lastly, we present a detailed information-theoretic analysis of the introduced objects. In particular, we quantify concentration of a given source-field in terms of the volume factors, thresholds, and active regions.
Archive for Rational Mechanics and Analysis – Springer Journals
Published: Jun 1, 2023
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