Turbulent flows are not uniformly multifractal

Abstract

The Frisch-Parisi multifractal formalism remains the most compelling rationalisation for anomalous scaling in fully developed turbulence. We now show that this formalism can be adapted locally to reveal the spatial distribution of generalized dimensions and of how multifractal the energy dissipation field is. In particular, we show that most regions of the flow are close to being mono-fractal and these are interspersed with islands of multifractality corresponding to the most singular structures in the flow. By defining a suitable measure $\Phi ({\bf x})$ of the spatial variation of multifractality, we show that this grows logarithmically with the extent to which the energy dissipation varies locally around ${\bf x}$. These results suggest ways to understand how singularities could arise in disparate regions of a flow and provides new directions in understanding anomalous dissipation and intermittency. We then employ the same technique to a non-intermittent, model turbulent flow to check the robustness of our conclusions.