Abstract
A statistical closure of the Navier–Stokes hierarchy which leads to equations for the two-point, two-time covariance of the velocity field for stationary, homogeneous isotropic turbulence is presented. It is a generalisation of the self-consistent field method due to Edwards (1964) for the stationary, single-time velocity covariance. The probability distribution functional is obtained, in the form of a series, from the Liouville equation by means of a perturbation expansion about a Gaussian distribution, which is chosen to give the exact two-point, two-time covariance. The triple moment is calculated in terms of an ensemble-averaged infinitesimal velocity-field propagator, and shown to yield the Edwards result as a special case. The use of a Gaussian zero-order distribution has been found to justify the introduction of a fluctuation-response relation, which is in accord with modern dynamical theories. In a sense this work completes the analogy drawn by Edwards between turbulence and Brownian motion. Originally Edwards had shown that the noise input was determined by the correlation of the velocity field with the externally applied stirring forces but was unable to determine the system response. Now we find that the system response is determined by the correlation of the velocity field with internal quasi-entropic forces. This analysis is valid to all orders of perturbation theory, and allows the recovery of the local energy transfer (LET) theory, which had previously been derived by more heuristical methods. The LET theory is known to be in good agreement with experimental results. It is also unique among two-point statistical closures in displaying an acceptable (i.e. non-Markovian) relationship between the transfer spectrum and the system response, in accordance with experimental results. As a result of the latter property, it is compatible with the Kolmogorov (K41) spectral phenomenology.
Highlights
It is sometimes remarked that the theory of turbulence is mired in controversy
We argue that the system response can be identified as the ensemble average of the Liouville operator against the Gaussian ground-state pdf and can be calculated as a generalised fluctuation response relation
First there is the question of the relevance of a fluctuation-response relation to turbulence, a subject which has received a certain amount of attention over the years
Summary
It is sometimes remarked that the theory of turbulence is mired in controversy. What we have is that perceptions of the subject appear to be quite static and dominated by a small number of issues which arose in the 1960s/70s. None of these issues is necessarily of any great significance and some have been effectively resolved. In this respect, the statistical theory is really just like the rest of fundamental research on turbulence. For some remarks on this aspect of turbulence research, see the review by Sreenivasan [1]
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