Hydrodynamic turbulence as a problem in nonequilibrium statistical mechanics

Abstract

The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems describing units of fluid of smaller and smaller spatial extent. These units are macroscopic but have a few degrees of freedom, and they can be studied by the methods of (microscopic) nonequilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specifically, we obtain the formula ζ(p)=p/3-1/Inκ In Γ(p/3 + 1) for the exponents of the structure functions (left angle bracket|Δ(r)V|(p) right angle bracket ~ r(ζ(p)). The meaning of the adjustable parameter κ is that when an eddy of size r has decayed to eddies of size r/κ, their energies have a thermal distribution. The above formula, with (In κ)⁻¹ = .32 ± .01 is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture that can thus also be used in related problems.