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

1977/08/01 by Dieter Forster, David R. Nelson, Michael J. Stephen · 108 citations
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Spectroscopy and Quantum Chemical Studies #stochastic dynamics and bifurcation

paper · doi:10.1103/physreva.16.732

openalex publication_date 1977/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

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, \ensuremath∼\frac(logt)^\ensuremath-(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 is unclear, the results apply to a forced Burger's equation in one dimension. A large class of models produces a spectral function E(k) which behaves as k2 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.

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