A modified Lin equation for the energy balance in isotropic turbulence

Abstract

• Resolution of the scale-invariance paradox is used to modify the Lin equation. • Exact symmetries justify the division of wavenumber space into different regions. • The roles of these different regions can be established by energy conservation. • Equivalent decompositions can be applied to the Navier–Stokes equations. • This could be the basis of new theoretical approaches to the turbulence problem. At sufficiently large Reynolds numbers, turbulence is expected to exhibit scale-invariance in an intermediate (“inertial”) range of wavenumbers, as shown by power law behavior of the energy spectrum and also by a constant rate of energy transfer through wavenumber. However, there is an apparent contradiction between the definition of the energy flux (i.e., the integral of the transfer spectrum) and the observed behavior of the transfer spectrum itself. This is because the transfer spectrum T ( k ) is invariably found to have a zero-crossing at a single point (at k = k * ), implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k = k * . This behavior was formulated as a paradox and resolved by the introduction of filtered/partitioned transfer spectra, which exploited the symmetries of the triadic interactions (J. Phys. A: Math. Theor., 2008). In this paper we consider the more general implications of that procedure for the spectral energy balance equation, also known as the Lin equation. It is argued that the resulting modified Lin equations (and their corresponding Navier–Stokes equations) offer a new starting point for both numerical and theoretical methods, which may lead to a better understanding of the underlying energy transfer processes in turbulence. In particular the filtered partitioned transfer spectra could provide a basis for a hybrid approach to the statistical closure problem, with the different spectra being tackled using different methods.