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Abstract
In order to describe the phenomenon of intermittency in wall turbulence and, more particularly, the behaviour of moments and and intermittency exponents ζP with the order p and distance to the wall, we developed a new geometrical framework called “entropic-skins geometry” based on the notion of scale-entropy which is here applied to an experimental database of boundary layer flows. Each moment has its own spatial multi-scale support Ωp (“skin”). The model assumes the existence of a hierarchy of multi-scale sets Ωp ranged from the “bulk” to the “crest”. The crest noted characterizes the geometrical support where the most intermittent (the highest) fluctuations in energy dissipation occur; the bulk is the geometrical support for the whole range of fluctuations. The model assumes then the existence of a dynamical flux through the hierarchy of skins. The specific case where skins display a fractal structure is investigated. Bulk fractal dimension and crest dimension are linked by a scale-entropy flux defining a reversibility efficiency (d is the embedding dimension). The model, initially developed for homogeneous and isotropic turbulent flows, is applied here to wall bounded turbulence where intermittency exponents are measured by extended self-similarity. We obtained for intermittency exponents the analytical expression with γ ≈ 0.36 in agreement with experimental results.
Highlights
We observe here clearly what is the main advantage provided by entropic skins geometry: each structure function has its own fractal dimension Δp and the whole set is linked by the intermittency efficiency γ = (Δ f − Δ ∞ ) / (d − Δ ∞ )
We developed a geometrical description whose main advantage is to link classical statistics based on structure functions and a multi-scale geometry lying on a hierarchy of fractal dimensions
In order to describe the phenomenon of intermittency in wall bounded turbulence such as a turbulent boundary layer, we introduced a new geometrical framework based on scale-entropy and a hierarchy of fractal sets linked to each other by scale-entropy dynamics which is characterized by intermittency efficiency
Summary
Kolmogorov’s theory of fully developed turbulence assumes that energy dissipation occurs through an homogenous and isotropic way [1,2]. Following Kolmogorov’s formalism, turbulent flows are p usually studied using structure functions δ V r or ε r p where δ V r is the velocity difference across a distance r and εr, the rate of energy dissipation per unit mass averaged over a ball of size r. Extended Self-Similarity and Intermittency Exponents for Homogenous Isotropic Turbulence. ESS represents an indirect determination of scaling exponents via measurements obtained at low and moderate Reynolds numbers in order to gather results that would constitute the limit at high Reynolds numbers. For homogeneous and isotropic turbulent flows (scaling exponents noted ζpHI), intermittency represents for low orders a slight deviation from the Kolmogorov’s theory; its effect on the spectrum of turbulence (order 2) is almost negligible but departure is important at large orders.
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