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 $15\le$ Taylor-Reynolds number $R{e}_{\lambda }\le 200$ up to $R{e}_{\lambda }\approx 45000$, employing a reduced wave vector set method (introduced earlier) to approximately solve the Navier-Stokes equation. First, also for these extremely high Reynolds numbers $R{e}_{\lambda }$, the energy spectra as well as the higher moments – when scaled by the spectral intensity at the wave number $k_p$ of peak dissipation – can be described by {\it one universal} function of $k\mathrm{/}{k}_p$ for all $R{e}_{\lambda }$. Second, the ISR scaling exponents ${\zeta }_m$ of this universal function are in agreement with the 1941 Kolmogorov theory (the better, the large $R{e}_{\lambda }$ is), as is the $R{e}_{\lambda }$ dependence of $k_p$. Only around $k_p$ viscous damping leads to slight energy pileup in the spectra, as in the experimental data (bottleneck phenomenon).