Research Papers

A New Extrapolation-Based Uncertainty Estimator for Computational Fluid Dynamics

[+] Author and Article Information
Tyrone S. Phillips

Department of Aerospace
and Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: tphilli6@gmail.com

Christopher J. Roy

Department of Aerospace
and Ocean Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: cjroy@vt.edu

Manuscript received December 2, 2014; final manuscript received December 22, 2016; published online January 20, 2017. Editor: Ashley F. Emery.

J. Verif. Valid. Uncert 1(4), 041006 (Jan 20, 2017) (13 pages) Paper No: VVUQ-14-1006; doi: 10.1115/1.4035666 History: Received December 02, 2014; Revised December 22, 2016

A new Richardson extrapolation-based uncertainty estimator is developed which utilizes a global order of accuracy. The most significant difference between the proposed uncertainty estimator (referred to as the global deviation uncertainty estimator) and others in the literature is that we compute uncertainty estimates at all cells/nodes in the domain regardless of the local convergence behavior (i.e., even if the local solution is oscillatory with grid refinement). Various metrics are used to quantitatively calibrate and evaluate the uncertainty estimator compared to the true solution. The metrics are used to assess the global deviation uncertainty estimator compared to other commonly used uncertainty estimators of the same type such as the original grid convergence index (GCI) and the factor of safety method. Four two-dimensional, steady, inviscid flow fields with exact solutions are used to calibrate the parameters in the proposed uncertainty estimator and make up about 30% of the total solution data set. The evaluation data set is composed of several additional steady, two-dimensional and three-dimensional solutions computed using different computational fluid dynamics codes with exact solutions including a zero pressure gradient turbulent flat plate with a well-defined numerical benchmark. All solutions are formally first- or second-order accurate. The global deviation uncertainty estimator is developed using an empirical approach with a focus on local variables and shows significant improvement compared to existing extrapolation-based uncertainty estimates, even when applied to regions where the local convergence behavior is divergent or oscillatory.

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Fig. 1

Factor of safety of the global deviation uncertainty estimator compared to the GCI factors of safety

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Fig. 2

Local effectivity index plotted versus the convergence ratio for all local quantities of the calibration data set

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Fig. 3

Comparison of the data for two different uncertainty estimators with quadratic fit and bounds for less scatter (top row) and more scatter (bottom row)

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Fig. 4

Pressure, streamlines, and sample grids for the calibration data set: (a) subsonic, (b) supersonic, (c) Ringleb’s, and (d) supersonic vortex

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Fig. 5

Sample Loci-CHEM grid topologies showing a (a) cartesian tetrahedral grid, (b) skewed, prismatic grid, (c) highly skewed, hexahedral grid, and (d) skewed, hybrid grid

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Fig. 6

Sample PARNASSOS computational grids showing a (a) cartesian grid, (b) stretched grid, and (c) skewed grid

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Fig. 7

Spalart-Allmaras 69 × 49 turbulent flat plate computational grid with the flat plate shown in at the bottom of the domain extending from zero to 2L

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Fig. 8

PDFs of inverse local effectivity index taken at different convergence ratios from Fig. 7

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Fig. 9

Comparison between non-normalized Δp (columns A and C) and normalized Δp (columns B and D) for error effectivity index and conservativeness: (a) p = pf, (b) p = pf, (c) p =  p̂, and (d) p =  p̂

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Fig. 10

Comparing the GCI with a factor of safety of 1.25 using three different methods of computing the observed order of accuracy

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Fig. 11

Comparison between possible discretization error estimators

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Fig. 12

Metrics for the GCI implementations and FS method using local quantities from the calibration data set

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Fig. 13

Proposed uncertainty estimator applied to the calibration data set

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Fig. 14

Evaluation of the global deviation uncertainty estimator using all of test data set

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Fig. 15

Turbulent boundary layer at L = 1 and corresponding uncertainty estimates for (a) the GCI using a factor of safety of 3.0, (b) the GCI using a factor of safety of 1.25, (c) the GCI using the globally averaged order of accuracy, (d) the factor of safety method, (e) and the global deviation uncertainty estimator



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