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Theory of turbulence
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1995 Rep. Prog. Phys. 58 1117
(http://iopscience./0034-4885/58/10/001)
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Rep. Prog. Phys. 58 (1995) 1117-l206. Printed in the UK
Theory of turbulence
W D b
Department of Physics and Astronomy, University of Edinburgh, Mayficld Road, Edinburgh EH9 312, UK
Abstract
This review is concerned with modem theoretical approaches to turbulence, in which
the problem can be seen as a branch of statistical field theory, and where the treatment
has been strongly influenced by analogies with the quantum many-body problem. The
dominant themes treated are the development (since the 1950s) of renormalized pertur-
bation theories (RPT) and, more recently, of renormalization group (RG) methods.
As fluid dynamics is rarely part of the physics curriculum, in section 1 we introduce
some background concepts in fluid dynamics, followed by a skeleton treatment of the
phenomenology of turbulence in section 2, taking flow through a straight pipe or a plane
channel as a representative example. In section 3, the general statistical formulation of
the problem is given, leading to a moment closure problem, which is analogous to the
well known BBGKY hierarchy, and to the Kolmogorov -5/3 power law, which is a
consequence of dimensional analysis. In section 4, we show how RPT have been used
to tackle the moment closure problem, distinguishing between those which pat-
ible with the Kolmogorov spectrum and those which are not. In section 5, we discuss
the use of RG to reduce the number of degrees of freedom in the numerical simulation
of the turbule