Theory of turbulence

This review is concerned with modern 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 perturbation 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 are compatible 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 turbulent equations of motion, while giving a clear statement of the technical problems which lie in the way of doing this. Lastly, the theories are discussed in section 6, in terms of their ability to meet the stated goals, as assessed by numerical computation and comparison with experiment.