Longitudinal and Transverse structure functions in high Reynolds-number turbulence

TL;DR

This paper derives relationships between longitudinal and transverse structure functions in high Reynolds-number turbulence, confirming differences in their scaling exponents and mapping their probability density functions using Mellin transforms.

Contribution

It introduces a simplified model linking longitudinal and transverse structure functions and analyzes their scaling behavior across a range of high Reynolds numbers.

Findings

Longitudinal and transverse structure functions have different scaling exponents.

A clear correspondence exists between their respective scaling ranges.

The difference in exponents appears independent of Reynolds number.

Abstract

Using exact relations between velocity structure functions (Hill, Hill and Boratav, and Yakhot) and neglecting pressure contributions in a first approximation, we obtain a closed system and derive simple order-dependent rescaling relationships between longitudinal and transverse structure functions. By means of numerical data with turbulent Reynolds numbers ranging from $\Re_\lambda=320$ to $\Re_\lambda=730$, we establish a clear correspondence between their respective scaling range, while confirming that their scaling exponents do differ. This difference does not seem to depend on Reynolds number. Making use of the Mellin transform, we further map longitudinal to (rescaled) transverse probability density functions.