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Abstract
A statistical closure of the NavierStokes hierarchy which leads to equations for the twopoint, twotime covariance of the velocity field for stationary, homogeneous isotropic turbulence is presented. It is a generalisation of the selfconsistent field method due to Edwards (1964) for the stationary, singletime velocity covariance. The probability distribution functional P [u, t] 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 twopoint, twotime covariance. The triple moment is calculated in terms of an ensembleaveraged infinitesimal velocityfield propagator, and shown to yield the Edwards result as a special case. The use of a Gaussian zeroorder distribution has been found to justify the introduction of a fluctuationresponse 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 quasientropic 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 twopoint statistical closures in displaying an acceptable (i.e. nonMarkovian) 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.
Original language  English 

Article number  375501 
Number of pages  35 
Journal  Journal of Physics A: Mathematical and Theoretical 
Volume  50 
Early online date  22 Aug 2017 
DOIs  
Publication status  Published  22 Aug 2017 
Keywords
 homogeneous isotropic turbulence
 local energy transfer theory
 field theory
 renormalised perturbation theory
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Dive into the research topics of 'A formal derivation of the local energy transfer (LET) theory of homogeneous turbulence'. Together they form a unique fingerprint.Projects
 2 Finished

Lab in a bubble
Jaroszynski, D., Boyd, M., Brunetti, E., Ersfeld, B., Hidding, B., McKenna, P., Noble, A., Sheng, Z., Vieux, G., Welsh, G. H. & Wiggins, M.
EPSRC (Engineering and Physical Sciences Research Council)
1/04/16 → 31/03/21
Project: Research

LaserlabEurope IV (H2020 INFRA IA)
Jaroszynski, D., Hidding, B., McKenna, P. & Sheng, Z.
European Commission  Horizon 2020
1/12/15 → 30/11/19
Project: Research
Research output
 7 Citations
 1 Doctoral Thesis

Investigation of the transfer and dissipation of energy in isotropic turbulence
Yoffe, S. R., 2 Apr 2012, Edinburgh. 292 p.Research output: Thesis › Doctoral Thesis