Energy stability analysis of turbulent incompressible flow based on the triple decomposition of the velocity gradient tensor

TL;DR

This paper conducts an energy stability analysis of turbulent incompressible flow using a triple decomposition of the velocity gradient tensor, revealing the evolution and stability characteristics of different flow structures at various scales.

Contribution

It introduces a refined energy stability analysis based on triple decomposition, providing new insights into the stability and evolution of flow structures in turbulence.

Findings

Unstable irrotational flow structures evolve into shear and rotational flows.

Viscous dissipation stabilizes flow structures near microscales.

Stable flow structures can persist over extended periods in turbulence.

Abstract

In the context of flow visualization a triple decomposition of the velocity gradient into irrotational straining flow, shear flow and rigid body rotational flow was proposed by Kolar in 2007 [V. Kolar, International journal of heat and fluid flow, 28, 638, (2007)], which has recently received renewed interest. The triple decomposition opens for a refined energy stability analysis of the Navier-Stokes equations, with implications for the mathematical analysis of the structure, computability and regularity of turbulent flow. We here perform an energy stability analysis of turbulent incompressible flow, which suggests a scenario where at macroscopic scales any exponentially unstable irrotational straining flow structures rapidly evolve towards linearly unstable shear flow and stable rigid body rotational flow. This scenario does not rule out irrotational straining flow close to the Kolmogorov microscales, since there viscous dissipation stabilizes the unstable flow structures. In contrast to worst case energy stability estimates, this refined stability analysis reflects the existence of stable flow structures in turbulence over extended time.