Renormalisation Group Methods Applied to the Numerical Simulation of Fluid Turbulence

Abstract

Renormalisation group (or RG) is a powerful new method of tackling physical problems involving many length or energy scales. Originally a procedure for removing the divergences of quantum electrodynamics, it has recently come to prominence in the study of critical phenomena. In the context of fluid turbulence (which is what interests us here) RGmay be seen (in principle, at least) as a systematic way of progressively eliminating the effect of the smallest eddies; then the next smallest eddies; and so on: and replacing their mean effect by an effective turbulent viscosity. In other words, the molecular kinematic viscosity of the fluid v_o becomes renormalised by the collective interaction ofthe turbulent eddies. Or, if we prefer to think in terms of ‘modes’ of wavenumber k(rather than ‘eddies’), then for all k < k_c we may re-place v_o by v(k), where v(k) represents the mean effect of modes k > k_c. Here k_c is some cut-off.