Large-distance and long-time properties of a randomly stirred fluid

Dynamic renormalization-group methods are used to study the large-distance, long-time behavior of velocity correlations generated by the Navier-Stokes equations for a randomly stirred, incompressible fluid. Different models are defined, corresponding to a variety of Gaussian random forces. One of the models describes a fluid near thermal equilibrium, and gives rise to the usual long-time tail phenomena. Apart from simplifying the derivation of the latter, our methods clearly establish their universality, their connection with Galilean invariance, and their analytic form in two dimensions, $\sim \frac{{\left(logt\right)}^{-\frac{1}{2}}}{t}$. Nontrivial behavior results when the model is formally continued below two dimensions. Although the physical interpretation of the Navier-Stokes equations below $d=2\mathrm{}$ is unclear, the results apply to a forced Burger's equation in one dimension. A large class of models produces a spectral function $E\left(k\right)\mathrm{}$ which behaves as $k^2$ in three dimensions, as expected on the basis of equipartition. However, nonlinear effects (which become significant below four dimensions) control the infrared properties of models which force the Navier-Stokes equations at zero wave number.