Parity-Dependent Scaling of Velocity-Gradient Correlations in Turbulence

TL;DR

This paper explores how parity influences the scaling of velocity-gradient correlations in turbulence, revealing fundamental organizational principles and linking intermittent structures to scaling exponents.

Contribution

It uncovers parity-dependent organization of higher-order velocity-gradient correlations and connects intermittent turbulence structures to their scaling behavior.

Findings

Odd-odd correlations scale as r^{-4/3} with weak order dependence

Even-even correlations have systematically different exponents

Parity under sign reversal organizes higher-order turbulent correlations

Abstract

We investigate two-point velocity-gradient correlation functions in homogeneous isotropic turbulence using exact relations and direct numerical simulations. The second-order gradient correlation is shown to be exactly related to the Laplacian of the velocity correlation, implying inertial-range scaling $C_2^{1,1}(r)\sim r^{-4/3}$. At higher orders, we uncover a parity-dependent organization of gradient correlations: odd-odd correlations exhibit scaling close to $r^{-4/3}$ with weak dependence on order, whereas even-even correlations display systematically different exponents. We show that this distinction originates from the sign structure of the gradient field: sign decorrelation suppresses intermittent contributions in odd-odd sectors, while even-even correlations retain them and remain sensitive to the spatial organization of intense structures. The measured even-even exponents are quantitatively consistent, across two Reynolds numbers, with independently measured box-counting dimensions of intermittent gradient structures. These results identify parity under sign reversal as a fundamental organizing principle for higher-order turbulent correlations and establish a direct connection between sparse intermittent geometry and scaling exponents in turbulence.