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Research Papers

A Contribution for the Assessment of Discretization Error Estimators Based on Grid Refinement Studies

[+] Author and Article Information
L. Eça

Mechanical Engineering Department,
Instituto Superior Técnico, ULisboa
Av. Rovisco Pais 1,
Lisbon 1049-001, Portugal
e-mail: luis.eca@ist.utl.pt

G. Vaz

Maritime Research Institute Netherlands,
CFD Development,
P.O. Box 28,
Wageningen 6700AA, The Netherlands
e-mail: G.Vaz@marin.nl

M. Hoekstra

Consultant,
Voorthuizen, The Netherlands

1Corresponding author.

2Naturally, one can manufacture exact solutions for turbulent flows [6].

Manuscript received December 14, 2017; final manuscript received July 3, 2018; published online August 6, 2018. Assoc. Editor: Christopher J. Roy.

J. Verif. Valid. Uncert 3(2), 021001 (Aug 06, 2018) (10 pages) Paper No: VVUQ-17-1036; doi: 10.1115/1.4040803 History: Received December 14, 2017; Revised July 03, 2018

Abstract

This paper presents grid refinement studies for statistically steady, two-dimensional (2D) flows of an incompressible fluid: a flat plate at Reynolds numbers equal to 107, 108, and 109 and the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg with Re = 6 × 106. Results are based on the numerical solution of the Reynolds-averaged Navier–Stokes (RANS) equations supplemented by one of three eddy-viscosity turbulence models of choice: the one-equation model of Spalart and Allmaras and the two-equation models k – ω SST and $k−kL$. Grid refinement studies are performed in sets of geometrically similar structured grids, permitting an unambiguous definition of the typical cell size, using double precision and an iterative convergence criterion that guarantees a numerical error dominated by the discretization error. For each case, different grid sets with the same number of cells but different near-wall spacings are used to generate a data set that allows more than one estimation of the numerical uncertainty for similar grid densities. The selected flow quantities include functional (integral), surface, and local flow quantities, namely, drag/resistance and lift coefficients; skin friction and pressure coefficients at the wall; and mean velocity components and eddy viscosity at specified locations in the boundary-layer region. An extra set of grids significantly more refined than those proposed for the estimation of the numerical uncertainty is generated for each test case. Using power law extrapolations, these extra solutions are used to obtain an approximation of the exact solution that allows the assessment of the performance of the numerical uncertainty estimations performed for the basis data set. However, it must be stated that with grids up to 2.5 (plate) and 8.46 (airfoil) million cells in two dimensions, the asymptotic range is not attained for many of the selected flow quantities. All this data is available online to the community.

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References

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Figures

Fig. 1

Computational domain for test cases I, II, and III that correspond to the flow over a flat plate

Fig. 2

Computational domain for test cases IV, V, and VI that correspond to the flow around the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg, respectively

Fig. 3

Selected interior locations for test cases IV, V, and VI that correspond to the flow around the NACA 0012 airfoil at angles of attack of 0, 4, and 10 deg, respectively

Fig. 4

Maximum and average values of the near-wall cell sizes in wall coordinates. Flow over a flat plate test cases I, II, and III: case I, Re = 107, case II, Re = 108, and case III, Re = 109.

Fig. 5

Illustration of grids in the vicinity of the NACA 0012 airfoil for test cases IV, V, and VI

Fig. 6

Maximum and average values of the near-wall cell sizes in wall coordinates. Flow around the NACA 0012 airfoil, test cases IV, V, and VI: : case IV, α = 0 deg, case V, α = 4 deg, and case VI, α = 10 deg.

Fig. 7

Convergence of functional quantities with grid refinement for the six test cases. Extrapolated solution to cell size zero is based on the solid line fitted to the open circles. Workshop data correspond to ri ≥ 1.

Fig. 8

Skin friction distribution for the flow over a flat plate at Re = 108 (case II) and skin friction and pressure coefficients distributions for the flow around the NACA 0012 with α = 10 deg (case VI). Results obtained in the finest grids, ri = 0.364 for case II and ri = 0.333 for case VI.

Fig. 9

Convergence of skin friction and pressure coefficients with grid refinement for the selected test cases. Extrapolated solution to cell size zero is based on the solid line fitted to the open circles. Workshop data correspond to ri ≥ 1.

Fig. 10

Examples of smooth convergence of Cartesian velocity components Vx, Vy and eddy viscosity νt with grid refinement at interior locations of the selected test cases. Extrapolated solution to cell size zero is based on the solid line fitted to the open circles. Workshop data correspond to ri ≥ 1.

Fig. 11

Examples of nonsmooth convergence of Cartesian velocity components Vx, Vy and eddy viscosity νt with grid refinement at interior locations of the selected test cases. Extrapolated solution to cell size zero is based on the solid line fitted to the open circles. Workshop data correspond to ri ≥ 1.

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