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

We examine the behavior of the Kolmogorov constants C2, Ck, and Ck1, 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, C2/Ck1 and Ck/Ck1, exhibit clear dependence on the microscale Reynolds number Rλ, implying that they cannot all be independent of Rλ. In particular, it is found that (Ck1/C2−0.25)=1.95Rλ−0.68. The study further reveals that the widely used relation C2=4.02Ck1 holds only asymptotically when Rλ≳105. It is also found that C2 has much stronger Rλ dependence than either Ck or Ck1 if the latter indeed has a systematic dependence on Rλ. We further show that the varying dependence on Rλ 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.