# A Scale-Entropy Diffusion Equation for Wall Turbulence

## Abstract

We applied on a database of PIV fields obtained at Laboratoire de Mécanique de Lille corresponding to a turbulent boundary layer the statistical and geometrical tools defined in the context of entropic-skins theory. We are interested by the spatial organization of velocity fluctuations. We define the absolute value of velocity fluctuation *δ* *V* defined relatively to the mean velocity. For given value *δ* *V* _{ s } (the threshold), the set *Ω*(*δ* *V* _{ s }) is defined by taking the points on the field where *δ* *V*≤*δ* *V* _{ s }. We thus define a hierarchy of sets for the threshold *δ* *V* _{ s } ranging from the Kolmogorov velocity (the corresponding set is noted *Ω* _{ K }) to the turbulent intensity *U*′ (the corresponding set is noted *Ω* _{ U′}). We then characterize the multi-scale features of the sets *Ω*(*δ* *V* _{ s }). It is shown that, between Taylor and integral scale, the set *Ω*(*δ* *V* _{ s }) can be considered as self-similar which fractal dimension is noted *D* _{ s }. We found that fractal dimension varies linearly with logarithm of ratio *δ* *V* _{ s }/*U*′. The relation is *D* _{ s }=2+*β*ln (*δ* *V* _{ s }/*U*′) with *β*≈0.12–0.26: this result is obtained for all the values *y* ^{+} we worked with. We then defined an equivalent dispersion scale *l* _{ e } such as \(N(\delta V_{s})-N_{K}=l_{e}^{2}\). It is shown that \(\delta V_{s}/U'\sim l_{e}^{1.52}\). We thus can write *D* _{ s }=2+*β*′ln (*l* _{ e }/*l* _{0}) with *β*′≈0.18–0.39. These results are interpreted in the context of a scale-entropy diffusion equation introduced to characterize multi-scale geometrical features of turbulence.

## Keywords

Fractal Dimension Turbulent Intensity Velocity Fluctuation Integral Scale Wall Turbulence## References

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