Estimating uncertainties in statistics computed from direct numerical simulation

Abstract

1 Rigorous assessment of uncertainty is crucial to the utility of 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 nontrivial 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 atdomain sizes are investigated. For each case, a sequence of meshes was generated by first designing a "nominal" mesh using standard heuristics for wall-bounded sim-

A methodology to evaluate statistical errors in DNS data of plane channel flows

“…Since statistical quantities are often provided as outputs of a DNS, the uncertainties associated to these statistics come from discretization errors and sampling errors. Recently, a discussion on this matter was conducted by Oliver et al [21] , where it was stated that is not common in the DNS literature to estimate the uncertainties of the statistical quantities. One of the main difficulties is related to the fact that samples used to generate DNS statistics are originated from a time history and spatial field that are generally not independent and the procedure used to mitigate this fact, namely taking samples far apart in time and space, does not eliminate it.…”

Reynolds-number dependence of turbulent skin-friction drag reduction induced by spanwise forcing

“…The standard uncertainties δC f,0 and δC f are evaluated with the procedure described by Oliver et al (2014), by calculating an integral timescale via an autoregressive method. This is slightly different from the strategy employed by Gatti & Quadrio (2013), who estimated the timescale differently, and it is interesting to note that the two methods end up with essentially the same result.…”

Evaluating large eddy simulation results based on error analysis

“…While such numerical and modeling errors have drawn the interest of many researchers, very little attention has been paid to other sources of uncertainties, such as insufficiently large computational domains, improper inflow conditions, finite initial transient or sampling errors interfering with results of LES [7,9,28,46]. Furthermore, reference data reported in the literature of the same configuration can differ significantly [63] from each other.…”

“…For further discussion, please refer to, e.g., [47]. In addition, it should be noted here, that in the case of modest sample size, it is advantageous to estimate ρ i (s) by fitting an autoregressive time series models to reduce spurious oscillations as described in [46]. By means of the central limit theorem, the standard deviation of the estimator of the mean velocity U i,mean can be determined as σ t U i,mean = σ t U i / √ N t .…”