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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 Re τ ≈ 180. Both small (L x /δ × L z /δ = 4π × 2π) and large (L x /δ × L z /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.

© 2014 AIP Publishing LLC

Received Wed Sep 25 00:00:00 UTC 2013
Accepted Wed Feb 05 00:00:00 UTC 2014
Published online Thu Mar 06 00:00:00 UTC 2014

Acknowledgments:
The work presented here was supported by the Department of Energy [National Nuclear Security Administration] under Award No. DE-FC52-08NA28615 and the National Science Foundation under Award No. OCI-0749223.

The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported here. Finally, the authors wish to thank Mr. Myoungkyu Lee for the use of his simulation code as well as his assistance in generating several of the DNS runs.

Article outline:

I. INTRODUCTION
II. METHODOLOGY
A. Sampling error
B. Discretization error
1. Assessing order of accuracy without sampling error
2. Accounting for sampling error
C. Illustrative example: The Lorenz equations
1. Sampling error estimator performance
2. Bayesian Richardson extrapolation results
III. DNS OF *Re* _{τ} = 180 CHANNEL FLOW
A. Discretization, sampling, and prior details
B. Small domain results
1. Centerline mean velocity
2. Skin friction
3. Summary of results for single-point statistics
C. Large domain results
1. Centerline mean velocity and skin friction
2. Summary of results for single-point statistics
IV. CONCLUSIONS

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2014-03-06

2016-12-10

### 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 Re τ ≈ 180. Both small (L x /δ × L z /δ = 4π × 2π) and large (L x /δ × L z /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.

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