Hydrodynamic fluctuations in the Kolmogorov flow: Linear regime

The Landau-Lifshitz fluctuating hydrodynamics is used to study the statisti cal properties of the linearized Kolmogorov flow. The relative simplicity o f this flow allows a detailed analysis of the fluctuation spectrum from nea r equilibrium regime up to the vicinity of the first convective instability threshold. It is shown that in the long time limit the flow behaves as an incompressible fluid, regardless of the value of the Reynolds number. This is not the case for the short time behavior where the incompressibility ass umption leads in general to a wrong form of the static correlation function s, except near the instability threshold. The theoretical predictions are c onfirmed by numerical simulations of the full nonlinear fluctuating hydrody namic equations. [S1063-651X(99)09905-5].