Generalized multifractal analysis of the mixing rate across turbulent jet scalar interfaces using the multiscale-minima meshless (M³) method

Abstract

This work demonstrates generalized multifractal analysis on scalar conditional dissipation rate fields using the multiscale-minima meshless (M³) method. We investigate how the generalized multifractal dimension varies as a function of scale in order to characterize the geometrical structure of the scalar dissipation regions. We also examine how the generalized multifractal dimension varies with the multifractal exponent. The investigation is carried out using a high-resolution experimental two-dimensional image database of fully developed turbulent scalar fields in jets. The conditional dissipation field on scalar interfaces, which represents the mixing rate across the interfaces, is calculated for the outer interface with Taylor's hypothesis in the jet similarity plane. Isosets of multifractal exponents of the dissipation field are identified. Examination of the generalized multifractal dimension of the isosets using the M³ method shows strong dependence on scale. This indicates the presence of scalar conditional dissipation regions whose geometrical structure varies from point-like at the smallest resolved scales, to effectively tube-like at intermediate scales, to effectively space-filling at the largest scales. Since spatially two-dimensional scalar slices of the spatially three-dimensional scalar field were used, this suggests that the scalar conditional dissipation structures in three-dimensional space would be such that they are tube-like at the smallest resolved scales, effectively sheet-like at intermediate scales and effectively space-filling at the largest scales. This suggests the presence of geometrically scale-dependent spiral-like structures and this is consistent with previous observations that the conditional signatures of such structures consist of striation patterns of the dissipation across scalar interfaces. The scale dependence of the generalized multifractal dimension is found to be steepest for the multifractal exponent corresponding to the peak of the probability density function of the multifractal exponents, i.e. for the most probable multifractal exponent. The present findings show that generalized multifractal analysis with the M³ method provides a tool for quantifying the geometrical structure across various scales, including large scales, as well as for capturing scale-dependent aspects of the conditional dissipation rate which are useful capabilities for modeling the mixing rate across turbulent interfaces.

Keywords:

• dissipation
• mixing
• diffusion
• fractals
• multiscale geometry

Acknowledgements

We thank Aaron P. Freeman, Jennifer Shockro, and Fazlul R. Zubair for useful discussions. We are especially grateful to the referees for their insightful comments.