On the Kolmogorov Constants for the Second-Order Structure Function and the Energy Spectrum

TL;DR

This paper investigates how Kolmogorov constants related to turbulence spectra depend on the Reynolds number, revealing their ratios vary with R_{ extlambda} and challenging some traditional assumptions in turbulence theory.

Contribution

It demonstrates the R_{ extlambda}-dependence of Kolmogorov constants and clarifies the conditions under which their classical relations hold, linking inertial range differences to these dependencies.

Findings

C_{k1}/C_2 - 0.25 = 1.95 R_{ extlambda}^{-0.68}

C_2 = 4.02 C_{k1} holds only for R_{ extlambda} <= 10^5

C_2 shows stronger R_{ extlambda} dependence than C_k or C_{k1}

Abstract

We examine the behavior of the Kolmogorov constants C_2, C_k, and C_{k1}, which are, respectively, the prefactors of the second order longitudinal structure function, the three dimensional and one-dimensional longitudinal energy spectrum in the inertial range. We show that their ratios, C_2/C_{k1} and C_k/C_{k1}, exhibit clear dependence on the micro-scale Reynolds number R_{\lambda}, implying that they cannot all be independent of R_{\lambda}. In particular, it is found that (C_{k1}/C_2-0.25) = 1.95R_{\lambda}^{-0.68}. The study further reveals that the widely-used relation C_2 = 4.02 C_{k1} holds only asymptotically when R_{\lambda} <= 10^5. It is also found that C_2 has much stronger R_{\lambda}-dependence than either C_k, or C_{k1} if the latter indeed has a systematic dependence on R_{\lambda}. We further show that the variable dependence on R_{\lambda} of these three numbers can be attributed to the difference of the inertial range in real- and wavenumber-space, with inertial range in real-space known to be much shorter than that in wavenumber space.