Exact Resummations in the Theory of Hydrodynamic Turbulence: 0. Line-Resummed Diagrammatic Perturbation Approach

Abstract

The lectures presented by one of us (IP) at the Les Houches summer school dealt with the scaling properties of high Reynolds number turbulence in fluid flows. The results presented are available in the literature and there is no real need to reproduce them here. Quite on the contrary, some of the basic tools of the field and theoretical techniques are not available in a pedagogical format, and it seems worthwhile to present them here for the benefit of the interested student. We begin with a detailed exposition of the naive perturbation theory for the ensemble averages of hydrodynamic observables (the mean velocity, the response functions and the correlation functions). The effective expansion parameter in such a theory is the Reynolds number (Re); one needs therefore to perform infinite resummations to change the effective expansion parameter. We present in detail the Dyson-Wyld line resummation which allows one to dress the propagators, and to change the effective expansion parameter from Re to O(1). Next we develop the ``dressed vertex" representation of the diagrammatic series. Lastly we discuss in full detail the path-integral formulation of the statistical theory of turbulence, and show that it is equivalent order by order to the Dyson-Wyld theory. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field.