Kolmogorov's Hypotheses and Eulerian Turbulence Theory

Abstract

It is argued that Eulerian formulations are intrinsically unsuited for deriving the Kolmogorov theory because low-order Eulerian moments do not express sufficiently well a statistical dependence of nonsimultaneous amplitudes that accompanies the convection of small spatial scales by large spatial scales. Illustration is made by applying the direct-interaction approximation and a related, higher Eulerian approximation to an idealized convection problem and to a modified Navier-Stokes equation. Convection effects of low wavenumbers on high wavenumbers are removed in the modified equation, and as a consequence the direct-interaction approximation for it yields the Kolmogorov spectrum. Low-order Lagrangian moments provide a promisingly more complete description of the convection of small spatial scales by large, and a search for satisfactory Lagrangian closure approxi-mations seems highly desirable.