Investigation of renormalization group methods for the numerical simulation of isotropic turbulence

INTRODUCTION Over the years, our research into turbulence at Edinburgh has concentrated on the application of renormalization methods to the prediction of the energy spectrum of isotropic turbulence. General discussions of this work will be found elsewhere (McComb 1990, 1995), while accounts of specific progress have been given previously in this conference series (McComb & Shanmugasundaram 1983, McComb, Filipiak, Roberts & Watt, 1991). From a practical point of view, the most promising development in this area is undoubtedly Renormalization Group or RG. If we work in the Fourier representation, in principle, this involves the progressive averaging out of high-wavenumber modes in bands, with rescaling at each step, until a fixed point is reached. The result is, in effect, a ‘subgrid model’ for large-eddy simulation. RG has enjoyed its successes in other areas of statistical physics. However, its application to turbulence faces several technical difficulties, which have to be circumvented by uncontrolled approximations. Indeed, in view of the deterministic nature of the Navier-Stokes equations, it is clear that the operation of averaging out the high-wavenumber modes while keeping the low-wavenumber modes constant, cannot be done rigorously and in itself can only be an approximation. With points like this in mind, we have recently adopted direct numerical simulation as a tool for probing the basic feasibility of using RG techniques to reduce the number of degrees of freedom requiring to be numerically simulated. In this paper, we present some of the first results of this approach. We begin by discussing the RG approach in detail.