Estimating uncertainties in statistics computed from direct numerical simulation

Abstract

Rigorous assessment of uncertainty is crucial to the utility of direct numerical simulation (DNS) results. Uncertainties in the computed statistics arise from two sources: finite statistical sampling and the discretization of the Navier–Stokes equations. Due to the presence of non-trivial sampling error, standard techniques for estimating discretization error (such as Richardson extrapolation) fail or are unreliable. This work provides a systematic and unified approach for estimating these errors. First, a sampling error estimator that accounts for correlation in the input data is developed. Then, this sampling error estimate is used as part of a Bayesian extension of Richardson extrapolation in order to characterize the discretization error. These methods are tested using the Lorenz equations and are shown to perform well. These techniques are then used to investigate the sampling and discretization errors in the DNS of a wall-bounded turbulent flow at Reτ ≈ 180. Both small (Lx/δ × Lz/δ = 4π × 2π) and large (Lx/δ × Lz/δ = 12π × 4π) domain sizes are investigated. For each case, a sequence of meshes was generated by first designing a “nominal” mesh using standard heuristics for wall-bounded simulations. These nominal meshes were then coarsened to generate a sequence of grid resolutions appropriate for the Bayesian Richardson extrapolation method. In addition, the small box case is computationally inexpensive enough to allow simulation on a finer mesh, enabling the results of the extrapolation to be validated in a weak sense. For both cases, it is found that while the sampling uncertainty is large enough to make the order of accuracy difficult to determine, the estimated discretization errors are quite small. This indicates that the commonly used heuristics provide adequate resolution for this class of problems. However, it is also found that, for some quantities, the discretization error is not small relative to sampling error, indicating that the conventional wisdom that sampling error dominates discretization error for this class of simulations needs to be reevaluated.