Detailed kinetic energy dynamics of turbulent fluids

Despite significant experimental, computational and theoretical advances, a complete general mathematical framework for the turbulent motion of fluids has yet to be put forth. Here we propose such a modelling framework, by establishing a set of coupled equations of motion corresponding to the fluid's turbulent kinetic energy density, which are derived from the Navier-Stokes equations for an incompressible fluid. This approach most notably involves an amplitude-projection of the whole turbulent flow field onto a complete set of basis functions spanning the domain volume, followed by a volume average to simplify the representation. The result is a detailed account of the kinetic energy exchange between the projected amplitudes across all length scales of turbulent fluctuation. The nonlinear convective term in the Navier-Stokes equation specifically determines the selection rules for amplitude exchange, and yields the Kolmogorov energy cascade.