Universality in fully developed turbulence

We extend the numerical simulations of She et al. [Phys. Rev. Lett. 70, 3251 (1993)] of highly turbulent flow with 15≤Taylor-Reynolds numbers ${\mathrm{Re}}_{\lambda }$≤200 up to ${\mathrm{Re}}_{\lambda }$≊45 000, employing a reduced wave vector set method (introduced earlier) to approximately solve the Navier-Stokes equation. First, also for these extremely high Reynolds numbers ${\mathrm{Re}}_{\lambda }$, the energy spectra as well as the higher moments—when scaled by the spectral intensity at the wave number ${\mathit{k}}_{\mathit{p}}$ of peak dissipation—can be described by one universal function of k/${\mathit{k}}_{\mathit{p}}$ for all ${\mathrm{Re}}_{\lambda }$. Second, the k-space inertial subrange scaling exponents ${\mathrm{\zeta }}_{\mathit{m}}$ of this universal function are in agreement with the 1941 Kolmogorov theory (the better, the larger ${\mathrm{Re}}_{\lambda }$ is), as is the ${\mathrm{Re}}_{\lambda }$ dependence of ${\mathit{k}}_{\mathit{p}}$. Only around ${\mathit{k}}_{\mathit{p}}$, viscous damping leads to a slight energy pileup in the spectra, as in the experimental data (bottleneck phenomenon).