Energy spectra of certain randomly-stirred fluids

The velocity correlations of an incompressible fluid governed by the Navier-Stokes equations are studied in steady states maintained by random-white-noise stirring forces with varying spatial correlations. The asymptotic properties of the long-wavelength fluctuations are deduced by field-renormalization-group techniques. The results of Forster, Nelson, and Stephen are recovered for the random-force spectra these authors discuss, and a Kolmogorov spectrum is obtained when the force correlations have equal strength at all wave numbers, that is, when the force correlations behave as ${k}^{\ensuremath{-}d}$ in $d$ dimensions and $d>2$. Although the derivation is valid to all orders in the anomalous dimension, it implicitly assumes that there is no crossover in operator dimensionality.