Statistics of incompressible hydrodynamic turbulence: an alternative approach

TL;DR

This paper applies a new form of the Kolmogorov-Monin relation to compute energy cascade rates in incompressible hydrodynamic turbulence, demonstrating that cascade rates can be directly obtained from two-point increments without assuming isotropy, and confirming consistency with classical laws.

Contribution

It introduces an alternative theoretical approach to compute turbulence cascade rates directly from velocity and Lamb vector increments, bypassing isotropy assumptions.

Findings

Cascade rate $ ext{varepsilon}$ matches classical 4/3 law results.

Cascade rate can be obtained from simple two-point increments.

Results are consistent across different spatial resolutions.

Abstract

Using a recent alternative form of the Kolmogorov-Monin exact relation for fully developed hydrodynamics (HD) turbulence, the incompressible energy cascade rate $\varepsilon$ is computed. Under this current theoretical framework, for three-dimensional (3D) freely decaying homogeneous turbulence, the statistical properties of the fluid velocity (${\bf u}$), vorticity ($\boldsymbol \omega= \boldsymbol\nabla \times {\bf u}$) and Lamb vector ($\boldsymbol{\cal L}= \boldsymbol \omega \times {\bf u})$ are numerically studied. For different spatial resolutions, the numerical results show that $\varepsilon$ can be obtained directly as the simple products of two-point increments of ${\bf u}$ and $\boldsymbol{\cal L}$, without the assumption of isotropy. Finally, the results for the largest spatial resolutions show a clear agreement with the cascade rates computed from the classical 4/3 law for isotropic homogeneous HD turbulence.