Expressing turbulent kinetic energy as coarse-grained enstrophy or strain deformations

In turbulent flows, the element of fluid gets deformed by chaotic motion due to the formation of sharp velocity gradients. A direct connection between the element of fluid stresses and the energy balance still remains elusive. Here, an exact identity of incompressible turbulence is derived linking the velocity gradient norm across the scales to the total kinetic energy. In the context of three dimensional (3D) homogeneous turbulence, this relation can be specialised obtaining the expression of total kinetic energy decomposed either in terms of fluid element deformations due to strain motion or via the resolved scale enstrophy. Applied to data from direct numerical simulations (DNS), the decomposition reveals that, beyond the scales dominated by the external forcing, extensional and contractile deformations account approximately for 53% and 40% of the kinetic energy of the associated scale saturating to these values at the small scales; the remaining 7% is carried by the sign-indefinite one. From both these two identities one can derive an exact expression of the omni-directional energy spectrum which is formulated via an ultraviolet-infrared mixing of scales. Data from DNS indicated that this formulation of the energy spectrum reproduces both analytically and numerically the power-law behaviour of the well known Kolmogorov spectral scaling.