# A Schur decomposition reveals the richness of structure in homogeneous, isotropic turbulence as a consequence of localised shear

## TL;DR

This paper uses a Schur decomposition to analyze homogeneous, isotropic turbulence, revealing that local shear interactions, rather than normal tensor properties, drive flow structures and vorticity alignment.

## Contribution

It introduces a novel additive decomposition of the velocity gradient tensor using the Schur transform, highlighting the role of non-normality in turbulence structure formation.

## Key findings

- Flow tends to form disc-like structures due to shear interactions.
- Vorticity alignment with strain tensor eigenvectors results from local shear.
- Normal tensor properties do not explain flow structure tendencies.

## Abstract

An improved understanding of turbulence is essential for the effective modelling and control of industrial and geophysical processes. Homogeneous, isotropic turbulence (HIT) is the archetypal field for developing turbulence physics theory. Based on the Schur transform, we introduce an additive decomposition of the velocity gradient tensor into a normal part (containing the eigenvalues) and a non-normal or shear-related tensor. We re-interrogate some key properties of HIT and show that the the tendency of the flow to form disc-like structures is not a property of the normal tensor; it emerges from an interaction with the non-normality. Also, the alignment between the vorticity vector and the second eigenvector of the strain tensor is another consequence of local shear processes.