  When reporting statistics from simulations of statistically stationary
chaotic phenomenon, it is important to verify that the simulations are
time-converged. This condition is connected with the statistical error or
number of digits with which statistics can be reliably reported. In this work
we consider homogeneous and isotropic turbulence as a model problem to
investigate statistical convergence over finite simulation times. Specifically,
we investigate the time integration requirements that allow meaningful
reporting of the statistical error associated with finiteness of the temporal
domain. We address two key questions: 1) What is the appropriate range of
sampling frequency in large eddy time units? and 2) How long should a
simulation be performed in terms of large eddy time so that the statistical
error could be reliably reported. Our results indicate that proper sampling
frequency is on the order of 10 large eddy time scale. More importantly, we
find that reliable reporting of statistical errors requires simulation
durations orders of magnitude longer than typically performed. Our observations
of sampling frequency for homogeneous isotropic turbulence are also shown to
hold in turbulent channel flow.
