Renormalisation Group Analysis of Turbulent Hydrodynamics

Turbulent hydrodynamics is characterised by universal scaling properties of its structure functions. The basic framework for investigations of these functions has been set by Kolmogorov in 1941. His predictions for the scaling exponents, however, deviate from the numbers found in experiments and numerical simulations. It is a challenge for theoretical physics to derive these deviations on the basis of the Navier-Stokes equations. The renormalization group is believed to be a very promising tool for the analysis of turbulent systems, but a derivation of the scaling properties of the structure functions has so far not been achieved. In this work, we recall the problems involved, present an approach in the framework of the exact renormalisation group to overcome them, and present first numerical results. 1. Introduction The theoretical understanding of hydrodynamical turbulence still represents one of the big challenges of theoretical physics. In his fundamental work on this subject, Kolmogorov [1] introduced structure functions, describing the moments of velocity differences in a fluid. Assuming scale independence in a certain range of distances, he predicted scaling behaviour of the structure functions, associated with certain classical scaling exponents. The numbers for the exponents found experimentally and later in numerical simulations deviate, however, significantly from their classical values. It is still one of the unsolved problems of classical physics to derive the scaling behaviour from first principles. It is generally accepted that the behaviour of an incompressible fluid is on a fundamental level described by the Navier-Stokes equations, expressing the conservation of momentum of fluid elements. It should therefore in principle be possible to deduce the scaling exponents on the basis of the Navier-Stokes equations. This has, however, not been achieved so far. A promising approach seems to be the Renormalization Group (RG), which aims to describe the dependence of the correlation functions of a given field theory on the scale on which the system is observed. Beginning with the work of Forster et al. [2], numerous attempts have been made to apply the various formulations of the RG to turbulent hydrodynamics, but until today the observables proposed by Kolmogorov could not be deduced in accordance with experiment. See, for example, [3, 4] for some work on the RG approach to turbulence. There are basically two different approaches to the RG, the “field theoretic” and the “exact” RG. They are related to each other, so that the distinction may