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 R_λ, implying that they cannot all be independent of R_λ. In particular, it is found that (Cₖ₁/C₂-0.25)=1.95R_λ^-0.68. The study further reveals that the widely used relation C₂=4.02Cₖ₁ holds only asymptotically when R_λ10⁵. It is also found that C₂ has much stronger R_λ dependence than either Cₖ or Cₖ₁ 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.