The multiscale-minima meshless (M³) method: A novel approach to level crossings and generalized fractals with applications to turbulent interfaces

Abstract

A purely meshless multiscale approach, denoted as the multiscale-minima meshless (M³) method, is presented which offers a new tool useful for studies of generalized level crossings and fractals in complex phenomena including turbulence. The M³ method has its roots in the largest-empty box method pioneered by the author based on generalized level crossings, but with the key new aspect that it does not require any type of box. The M³ method enables a purely meshless evaluation of the generalized fractal dimension as a function of scale, i.e. in a completely grid-free manner without the use of boxes required in box-based methods, and offers an alternative tool to other meshless methods such as the wavelet-decomposition method and the caliper method. The key idea of the present M³ approach is the consideration of the shortest distance, or generalized level crossing scale, from each location within a reference region to the nearest turbulent feature of interest. This provides a new definition of the generalized fractal dimension. The theoretical formulation of the M³ method is presented showing that the shortest-distance probability density function provides a definition of, and a one-to-one correspondence to, a new generalized fractal dimension as a function of scale in multi-dimensional space. Computational validation results are presented comparing the M³ method and the box-counting method. As an experimental example, the utility of the M³ method for turbulence studies is demonstrated on a high-resolution experimental database of scalar interfaces in fully-developed turbulent jets. The ability of the M³ method to utilize physically-based reference regions, such as the region of fluid contained within the turbulent interface instead of artificial box regions, is demonstrated with multiscale-minima clusters observed. For the generalized fractal dimension function, scale dependence is found at both large scales and small scales. For the probability density function of shortest-distance scales, strong scale dependence is found at large scales and self-similar behavior at small scales with a scaling exponent that is very close to the theoretically proposed value of Mandelbrot for Kolmogorov scaling of power-law variances. These results show that finite-size effects have a significant effect on the apparent fractal dimension, to the extent that it exhibits no discernible constant value for the present data. At the same time, the results show the advantage of the scaling exponent of the shortest-distance probability density function in that it is able to reveal scaling even in the presence of finite-size effects. In addition to the capabilities presented, the proposed M³ approach contains conceptual ingredients that can be used in studies of large-scale versus small-scale effects, spacetime properties, generalized multifractals, angular anisotropic distributions, multiple generalized level crossings and related notions.

Keywords:

• chaos and fractals
• passive scalar turbulence
• turbulent mixing

Acknowledgements

This work is part of a research program on turbulence and the dynamics of flows facilitated by the National Science Foundation Career Program. Discussions with S. Piatrovich, F. Zubair, A. Freeman and J. Shockro are gratefully acknowledged. The insightful comments by the referees are gratefully acknowledged and in particular the referees are thanked for their suggestions to include analytical examples demonstrating the presented theory.