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

In the context of flow visualization a triple decomposition of the velocity gradient was proposed by Kolat in 2007 who demonstrated the technique in 2D, which was later realized in 3D by Nagata et al. in 2020. 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 any exponentially unstable irrotational flow structures rapidly evolve towards linearly unstable shear flow and stable rigid body rotational flow, which dissipates to heat. In contrast to worst case energy stability estimates, this refined stability analysis reflects the existence of stable flow structures in turbulence over extended time, and the robustness of average quantities.