Dynamic Triad Interactions and the Evolution of Turbulent Power Spectra
The Navier-Stokes equation is generally accepted to describe all aspects of the momentum balance of fluid flows – both laminar and turbulent. However, due to the nonlinearity of the equation and the wide range of scales interacting through the nonlinear term, the Navier-Stokes equation is notoriously difficult to solve, especially for turbulent flows at large Reynolds numbers. Herein, we therefore investigate a time-dependent solution in one primary spatial dimension to the 4-dimensional Navier-Stokes equation, representing the momentum balance for the instantaneous fluid convection velocity. The solution retains all terms in the Navier-Stokes equation, including both pressure and dissipation. The purpose is primarily to learn about the peculiar effects of the nonlinearity of the Navier-Stokes equation in all 4 dimensions, by presenting some numerical calculations of the flow development with different representative (including measured) input velocity records. Including the temporal component in the decomposition results in the addition of a temporal frequency contribution in the triad interaction matching condition. The solution reveals the dynamic development of non-equilibrium turbulence towards equilibrium, where a soliton-like behavior is eventually attained. The numerical model is compared to experiments and analysis. The classical picture, as described by Kolmogorov 1941 and Richardson, is discussed in light of these numerical, experimental and theoretical developments.
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