High-resolution direct numerical simulations (DNSs) of incompressible homogeneous turbulence in a periodic box with up to 4096(3) grid points were performed on the Earth Simulator computing system. DNS databases, including the present results, suggest that the normalized mean energy dissipation rate per unit mass tends to a constant, independent of the fluid kinematic viscosity nu as nu-->0. The DNS results also suggest that the energy spectrum in the inertial subrange almost follows the Kolmogorov k(-5/3) scaling law, where k is the wavenumber, but the exponent is steeper than -5/3 by about 0.1. (C) 2003 American Institute of Physics.