  Inspection of available data on the decay exponent for the kinetic energy of
homogeneous and isotropic turbulence (HIT) shows that it varies by as much as
100\%. Measurements and simulations often show no correspondence with
theoretical arguments, which are themselves varied. This situation is
unsatisfactory given that HIT is a building block of turbulence theory and
modeling. We take recourse to a large base of direct numerical simulations and
study decaying HIT for a variety of initial conditions. We show that the
Kolmogorov decay exponent and the Birkhoff-Saffman decay are both readily
observed, albeit approximately, for long periods of time if the initial
conditions are appropriately arranged. We also present, for both cases, other
turbulent statistics such as the velocity derivative skewness, energy spectra
and dissipation, and show that the decay and growth laws are approximately as
expected theoretically, though the wavenumber spectrum near the origin begins
to change relatively quickly, suggesting that the invariants do not strictly
exist. We comment briefly on why the decay exponent has varied so widely in
past experiments and simulations.
