2026/07/16 by Erik Bertram, E. Bertram
Engineering · Mathematics · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Micro and Nano Robotics
paper · pdf · doi:10.1016/j.physd.2026.135338
We develop a deterministic, scale-resolved formulation of energy transfer in the three-dimensional incompressible Navier–Stokes equations based on an explicit triadic decomposition of the nonlinear term in Fourier space. Using a systematic dyadic localization of the velocity field, we derive an exact representation of the nonlinear energy flux across scales and organize it in terms of interactions between well-defined scale components. Under suitable smoothness assumptions, we obtain an absolutely convergent triadic expansion and quantitative bounds that distinguish local and nonlocal contributions in scale space. This framework provides a transparent and fully explicit description of how energy transfer is mediated by triadic interactions and how scale locality emerges as a structural property of the nonlinearity. Building on this formulation, we revisit the classical inertial-range picture of turbulence from a deterministic perspective. We show that, under a scale-invariant flux assumption, the Kolmogorov − 5 / 3 scaling is formally consistent with the triadic energy-transfer mechanism at a structural level. The result does not rely on statistical assumptions, but instead follows from the structural properties of the Navier–Stokes equations combined with a scale-resolved representation of the energy flux. The present work thus provides a coherent synthesis of triadic interaction analysis, dyadic scale decomposition, and classical turbulence phenomenology, offering a deterministic framework that clarifies how Kolmogorov-type scaling constraints arise in the scale-resolved structure of the underlying equations.