Research Papers: Techniques and Procedures

Richardson Extrapolation-Based Discretization Uncertainty Estimation 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@vt.edu

Christopher J. Roy

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

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received August 19, 2013; final manuscript received April 1, 2014; published online September 10, 2014. Assoc. Editor: Zvi Rusak.

J. Fluids Eng 136(12), 121401 (Sep 10, 2014) (10 pages) Paper No: FE-13-1505; doi: 10.1115/1.4027353 History: Received August 19, 2013; Revised April 01, 2014

This study investigates the accuracy of various Richardson extrapolation-based discretization error and uncertainty estimators for problems in computational fluid dynamics (CFD). Richardson extrapolation uses two solutions on systematically refined grids to estimate the exact solution to the partial differential equations (PDEs) and is accurate only in the asymptotic range (i.e., when the grids are sufficiently fine). The uncertainty estimators investigated are variations of the grid convergence index and include a globally averaged observed order of accuracy, the factor of safety method, the correction factor method, and least-squares methods. Several 2D and 3D applications to the Euler, Navier–Stokes, and Reynolds-Averaged Navier–Stokes (RANS) with exact solutions and a 2D turbulent flat plate with a numerical benchmark are used to evaluate the uncertainty estimators. Local solution quantities (e.g., density, velocity, and pressure) have much slower grid convergence on coarser meshes than global quantities, resulting in nonasymptotic solutions and inaccurate Richardson extrapolation error estimates; however, an uncertainty estimate may still be required. The uncertainty estimators are applied to local solution quantities to evaluate accuracy for all possible types of convergence rates. Extensions were added where necessary for treatment of cases where the local convergence rate is oscillatory or divergent. The conservativeness and effectivity of the discretization uncertainty estimators are used to assess the relative merits of the different approaches.

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Fig. 1

Comparison of 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. 2

PARNASSOS Cartesian, stretched, and nonorthogonal grids

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Fig. 3

Samples of the Loci-CHEM computational grids showing the (a) Cartesian tetrahedral grid, (b) curvilinear prismatic grid, (c) highly skewed curvilinear hexahedral grid, and (d) curvilinear hybrid grid

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Fig. 4

Turbulent flat plate setup and the 69 × 49 grid

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Fig. 5

Comparison of the effectivity index for Richardson extrapolation versus various distance measures

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Fig. 6

Error effectivity indices for discretization error estimators

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Fig. 7

Uncertainty effectivity index and conservativeness for discretization uncertainty estimators




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