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

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 Kolář in 2007 [V. Kolář, “Vortex identification: New requirements and limitations,” Int. J. Heat Fluid Flow, 28, 638–652 (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 toward 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.