A Scale-Entropy Diffusion Equation for Wall Turbulence

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AbstractWe 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+βln (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.KeywordsFractal DimensionTurbulent IntensityVelocity FluctuationIntegral ScaleWall TurbulenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.