On the convergence of statistics in simulations of stationary turbulent flows

When performing computational simulation of any statistically stationary chaotic phenomenon, before reporting statistics, it is important to ensure that the simulations are time-converged. This condition is needed for rigorous reporting of the mean quantities by allowing a fair estimation of statistical error. 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 present our results in the context of the law of large numbers and central limit theorem. Given these two consideration we address two key questions: 1) How long should a simulation be performed in terms of large eddy time so that sufficient data samples are collected? 2) What is the appropriate range of sampling frequency in large eddy time units?