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+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.