Multiresolution robustness of the probability density of scalar fields and multiscale aspects of scalar interfaces in fully-developed turbulent jets

Abstract

This work focuses on the probability density function (pdf) of turbulent jet scalar fields with emphasis on the extent to which the pdf depends on resolution scale effects. The interpretation of the scalar pdf in terms of geometrical and physical properties of the turbulent scalar interfaces is utilized. The scalar pdf is examined as the product of the interfacial area–volume ratio, or perimeter–area ratio for two-dimensional data, and the interfacial thickness quantified as the averaged inverse magnitude of the scalar gradient across each interface. Because the area–volume ratio is expected to decrease with reduced resolution, whereas the interfacial thickness is expected to increase with reduced resolution, the possibility arises that the scalar pdf may exhibit only weak sensitivity to resolution scale effects. We demonstrate and investigate this expectation using a high-resolution experimental database of fully-developed turbulent scalar fields and interfaces in jets. Coarse graining is conducted by a two-step method consisting of applying an averaging filter and performing quadratic B-spline interpolation. Examination of resolution scale effects on the coarse-grained scalar pdf, along with the corresponding coarse-grained interfacial properties, shows that the scalar pdf exhibits significant robustness over a wide range of resolution scales as well as scalar thresholds. This finding of robustness, in jets, can be understood in terms of the multiscale interfacial properties. Specifically, the observed robustness arises from the dual self-similarity of the interfacial perimeter and thickness which are found to exhibit two self-similarity exponents of opposite signs and approximately equal magnitudes for the present jet conditions. In other flows, the degree of the multiresolution robustness can be expected to depend on the extent of the dual self-similarity and the relative magnitudes of the corresponding exponents.

Keywords:

• Turbulence
• Fluid interfaces
• Fractals
• Self-similarity
• Coarse graining

Acknowledgments

This work is a part of a research program on turbulence and the dynamics of flows. The authors acknowledge helpful informal discussions with Professor Emmanuel Villermaux regarding the idea of the possibility of robustness of turbulent scalar probability density functions to resolution scale effects. In addition, the authors acknowledge the insightful comments of the Referees especially regarding the use of the averaging filter in the coarse-graining procedure and the role of the self-similarity exponents in the pdf robustness.