Energy spectrum in high Reynolds number turbulence - high resolution DNS results

The energy spectrum and energy flux in high Reynolds number ($Re$) forced incompressible turbulence are investigated by using high-resolution DNS in a periodic box. We used negative viscosity (at a wavenumber range $kL<3$) to keep the total energy constant, and used well-developed turbulence fields as the initial conditions ($L$ is the integral length scale). The DNS with up to $6144^3$ grid points show that, after a transient period of the order of eddy turnover time, the standard deviation of the energy spectrum and that of the energy flux are largest at $kL\sim1$ and is an algebraically decreasing function of $kL$. As in previous studies, the energy spectra are insensitive to the values of $k_ {max}\eta$ when $k_{max}\eta\ge1$ ($\eta$ is the Kolmogorov length scale). The time-averaged, normalized energy spectra of high $Re$ turbulence at high $k$ overlap well with each other when they are plotted against $k\eta$. The normalized spectra have a slope steeper than -5/3 (the Kolmogorov scaling law) by factor 0.1 at $k\lambda\sim1$ ($\lambda$ is Taylor micro-scale). The DNS suggest that there is another wavenumber range ($k\lambda<1$), in which the spectrum has a slope close to -5/3, and also that the latter range increases with $Re$ and the Kolmogorov constant is $1.8\pm0.1$.