Abstract
The problem of calculating the energy spectrum of turbulent velocity pulsations in the case of homogeneous isotropic and stationary turbulence is considered. The domain of turbulent energy production is treated as “a black box” on which boundary the spectral energy flux is given. It is assumed that the spectrum is formatted due to intermodal interactions being local in the wave-number space that leads to a cascade mechanism of energy transfer along the wave-number spectrum and the possibility of using the renormalization-group method related to the Markovian features of the process under consideration. The obtained formula for energy spectrum is valid in a wide wave-number range and at arbitrary values of fluid viscosity. It is shown that in functional formulation of the statistical theory of turbulence, the procedure of separating local intermodal interactions, which govern energy transfer (straining effect), and filtering out nonlocal interactions, which have no influence on energy transfer (sweeping effect), is directly described without providing additional arguments or conjectures commonly used in the renormalization-group analysis of turbulent spectra.
Highlights
It is assumed that the spectrum is formatted due to intermodal interactions being local in the wave-number space that leads to a cascade mechanism of energy transfer along the wave-number spectrum and the possibility of using the renormalization-group method related to the Markovian features of the process under consideration
It is shown that in functional formulation of the statistical theory of turbulence, the procedure of separating local intermodal interactions, which govern energy transfer, and filtering out nonlocal interactions, which have no influence on energy transfer, is directly described without providing additional arguments or conjectures commonly used in the renormalization-group analysis of turbulent spectra
The specific feature of the system under consideration consists in the fact that in a fully developed turbulence, a large number of various mode scales are excited and the effect of modes of all scales appears to be essential and has to be accounted for since a single act of intermodal interactions is only a link in a long cascade chain via the mechanism of energy transfer from the range of large-scale modes, where the turbulent energy of stochastic fluid velocity pulsations due to a development of instability of large-scale flows is produced, into the range of small-scale modes where the energy dissipates due to fluid viscosity
Summary
The problem of calculating the energy distribution over wave-numbers of turbulent fluid (spectral energy density) is a subject of numerous investigations. 2) The inertial range is a domain of wave numbers where there is no energy production and the dissipation effects are negligible. In this range the spectral flux remains constant, and it goes the process of energy transfer from the range of small wave numbers into the range of large wave numbers via the cascade sequence of local intermodal interaction acts. We consider the problem of building the model that is true beyond the inertial range and obtain the formula for spectral energy distribution E (k ) with account for viscosity that is valid for all wave-numbers with the exception of ones from the energy production range
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