Kolmogorov constants for the second-order structure function and the energy spectrum

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 and 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 microscale Reynolds number R(λ), implying that they cannot all be independent of R(λ). In particular, it is found that (C(k1)/C(2)-0.25)=1.95R(λ)(-0.68). The study further reveals that the widely used relation C(2)=4.02C(k1) holds only asymptotically when R(λ)>/~10(5). It is also found that C(2) has much stronger R(λ) dependence than either C(k) or C(k1) if the latter indeed has a systematic dependence on R(λ). We further show that the varying dependence on R(λ) of these three numbers can be attributed to the difference of the inertial range in real- and wave-number space, with the inertial range in real-space known to be much shorter than that in wave-number space.