Turbulence DNS with up to 4096^3 grid points

High resolution direct numerical simulations (DNSs) of incompressible homogeneous turbulence in a periodic box with the number of the grid points up to 4096^3 and Taylor microscale Reynolds number about 1131 were performed on the Earth Simulator, with sustained performance up to 16.4 TFlops. The DNSs are based on a spectral method free from alias error. The numbers of the degree of freedom and the nonlinear interactions in the 4096^3-DNS are about M =40963x4 ~2.75 ×1011 , and M2 ~7.6 ×1022, respectively.
The talk will present some of the results of the DNS data analysis, with special emphasis on the Reynolds number dependence of the statistics including

1. the inertial subrange energy spectrum, and the dissipation range spectrum
2. the normalized mean energy dissipation rate, and
3. probability distribution function of velocity gradients, moments of velocity gradients including skewness and flatness factor, and the squares of the energy dissipation rate, enstrophy and pressure gradients.

1. the statistics of energy transfer
2. quantitative examination of Kolmogorv's log-normal hypothesis, and
3. eddy viscosity, in the inertial subrange