On the convergence of statistics in simulations of stationary turbulent flows

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.