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The purpose of this brief comunication is to improve a hypothesis of theprevious work of the author (de Divitiis, Theor Comput Fluid Dyn,doi:10.1007/s00162-010-0211-9) dealing with the finite--scale Lyapunov analysisof isotropic turbulence. There, the analytical expression of the structurefunction of the longitudinal velocity difference $\Delta u_r$ is derivedthrough a statistical analysis of the Fourier transformed Navier-Stokesequations, and by means of considerations regarding the scales of the velocityfluctuations, which arise from the Kolmogorov theory. Due to these latterconsiderations, this Lyapunov analysis seems to need some of the results of theKolmogorov theory.  This work proposes a more rigorous demonstration which leads to the samestructure function, without using the Kolmogorov scale. This proof assumes thatpair and triple longitudinal correlations are sufficient to determine thestatistics of $\Delta u_r$, and adopts a reasonable canonical decomposition ofthe velocity difference in terms of proper stochastic variables which areadequate to describe the mechanism of kinetic energy cascade.

Refinement of a previous hypothesis of the Lyapunov analysis of isotropic turbulence