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

In turbulent flows, the fluid element 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 with 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 deformations due to strain motion or via the resolved-scale enstrophy of the fluid element. Applied to data from direct numerical simulations (DNS) describing homogeneous and isotropic turbulence, the decomposition reveals that, beyond the scales dominated by the external forcing, contractile and extensional deformations account approximately for 55% and 40% of the kinetic energy of the associated scale while less than the remaining 5% is carried by the indefinite-type stresses. From these two identities one can derive an exact expression for the kinetic energy spectrum which is based solely on real space quantities providing a characterisation of the Kolomogorov constant as well. Numerical evidences show that this formulation of the energy spectrum reproduces the power-law behaviour of the Kolmogorov spectral scaling.