Re-examination of the infra-red properties of randomly stirred hydrodynamics

Dynamic renormalization group (RG) methods were originally used by Forster, Nelson and Stephen (FNS) to study the large-scale behaviour of randomly-stirred, incompressible fluids governed by the Navier-Stokes equations. Similar calculations using a variety of methods have been performed since, but have led to a discrepancy in results. In this paper, we carefully re-examine in $d$-dimensions the approaches used to calculate the renormalized viscosity increment and, by including an additional constraint which is neglected in many procedures, conclude that the original result of FNS is correct. By explicitly using step functions to control the domain of integration, we calculate a non-zero correction caused by boundary terms which cannot be ignored. We then go on to analyze how the noise renormalization, absent in many approaches, contributes an ${\mathcal O}(k^2)$ correction to the force autocorrelation and show conditions for this to be taken as a renormalization of the noise coefficient. Following this, we discuss the applicability of this RG procedure to the calculation of the inertial range properties of fluid turbulence.