A factor of safety method for quantitative estimates of grid-spacing and time-step uncertainties for solution verification is developed. It removes the two deficiencies of the grid convergence index and correction factor methods, namely, unreasonably small uncertainty when the estimated order of accuracy using the Richardson extrapolation method is greater than the theoretical order of accuracy and lack of statistical evidence that the interval of uncertainty at the 95% confidence level bounds the comparison error. Different error estimates are evaluated using the effectivity index. The uncertainty estimate builds on the correction factor method, but with significant improvements. The ratio of the estimated order of accuracy and theoretical order of accuracy P instead of the correction factor is used as the distance metric to the asymptotic range. The best error estimate is used to construct the uncertainty estimate. The assumption that the factor of safety is symmetric with respect to the asymptotic range was removed through the use of three instead of two factor of safety coefficients. The factor of safety method is validated using statistical analysis of 25 samples with different sizes based on 17 studies covering fluids, thermal, and structure disciplines. Only the factor of safety method, compared with the grid convergence index and correction factor methods, provides a reliability larger than 95% and a lower confidence limit greater than or equal to 1.2 at the 95% confidence level for the true mean of the parent population of the actual factor of safety. This conclusion is true for different studies, variables, ranges of P values, and single P values where multiple actual factors of safety are available. The number of samples is large and the range of P values is wide such that the factor of safety method is also valid for other applications including results not in the asymptotic range, which is typical in industrial and fluid engineering applications. An example for ship hydrodynamics is provided.

1.
Roache
,
P. J.
, 1998,
Verification and Validation in Computational Science and Engineering
,
Hermosa
,
Albuquerque, NM
.
2.
Celik
,
I. B.
,
Ghia
,
U.
,
Roache
,
P. J.
,
Freitas
,
C. J.
,
Coleman
,
H. W.
, and
Raad
,
P. E.
, 2008, “
Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications
,”
ASME J. Fluids Eng.
0098-2202,
130
(
7
), pp.
078001
.
3.
Cosner
,
R. R.
,
Oberkampf
,
W. L.
,
Rumsey
,
C. L.
,
Rahaim
,
C. P.
, and
Shih
,
T. I.-P.
, 2006, “
AIAA Committee on Standards for Computational Fluid Dynamics: Status and Plans
,”
44th Aerospace Sciences Meeting and Exhibit
, Reno, NV, Jan. 9–12, AIAA Paper No. 2006-889.
4.
Stern
,
F.
,
Wilson
,
R. V.
,
Coleman
,
H. W.
, and
Paterson
,
E. G.
, 2001, “
Comprehensive Approach to Verification and Validation of CFD Simulations—Part 1: Methodology and Procedures
,”
ASME J. Fluids Eng.
0098-2202,
123
(
4
), pp.
793
802
.
5.
Wilson
,
R. V.
,
Shao
,
J.
, and
Stern
,
F.
, 2004, “
Discussion: Criticisms of the ‘Correction Factor’ Verification Method (Roache, P. J., 2003, ASME J. Fluids Eng., 125(4), pp. 732–733)
,”
ASME J. Fluids Eng.
0098-2202,
126
(
4
), pp.
704
706
.
6.
Stern
,
F.
,
Wilson
,
R. V.
,
Coleman
,
H. W.
, and
Paterson
,
E. G.
, 1999, “
Verification and Validation of CFD Simulations
,” IIHR Report No. 407.
7.
Eça
,
L.
, and
Hoekstra
,
M.
, 2002, “
An Evaluation of Verification Procedures for CFD Applications
,”
24th Symposium on Naval Hydrodynamics
, Fukuoka, Japan, Jul. 8–13.
8.
Logan
,
R. W.
, and
Nitta
,
C. K.
, 2006, “
Comparing 10 Methods for Solution Verification, and Linking to Model Validation
,”
J. Aerosp. Comput. Inf. Commun.
,
3
(
7
), pp.
354
373
.
9.
Celik
,
I.
, and
Hu
,
G. S.
, 2004, “
Single Grid Error Estimation Using Error Transport Equation
,”
ASME J. Fluids Eng.
0098-2202,
126
(
5
), pp.
778
790
.
10.
Cavallo
,
P. A.
, and
Sinha
,
N.
, 2007, “
Error Quantification for Computational Aerodynamics Using an Error Transport Equation
,”
J. Aircr.
0021-8669,
44
(
6
), pp.
1954
1963
.
11.
Cavallo
,
P. A.
,
Sinha
,
N.
, and
O’Gara
,
M. R.
, 2008, “
Viscous Error Transport Equation for Error Quantification of Turbulent Flows
,”
38th Fluid Dynamics Conference and Exhibition
, Seattle, WA, Jun. 23–26, AIAA Paper No. 2008-3851, pp.
1
16
.
12.
ASME Performance Test Codes Committee PTC 61
, 2008, “
V&V 20: Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer
.”
13.
Eça
,
L.
, and
Hoekstra
,
M.
, 2006, “
Discretization Uncertainty Estimation Based on a Least Squares Version of the Grid Convergence Index
,”
Proceedings of the Second Workshop on CFD Uncertainty Analysis
, Instituto Superior Tecnico, Lisbon, Oct.
14.
Rumsey
,
C. L.
, and
Thomas
,
J. L.
, 2008, “
Application of FUN3D and CFL3D to the Third Workshop on CFD Uncertainty Analysis
,”
NASA
Report No. TM-2008-215537.
15.
2004,
Proceedings of the Workshop on CFD Uncertainty Analysis
,
L.
Eça
and
M.
Hoekstra
, eds., Lisbon, Portugal, Oct.
16.
Roache
,
P. J.
, 2009, private communication.
17.
Xing
,
T.
, and
Stern
,
F.
, 2008, “
Factors of Safety for Richardson Extrapolation for Industrial Applications
,” IIHR Report No. 466.
18.
Xing
,
T.
, and
Stern
,
F.
, 2009, “
Factors of Safety for Richardson Extrapolation for Industrial Applications
,” IIHR Report No. 469.
19.
Larsson
,
L.
,
Stern
,
F.
, and
Bertram
,
V.
, 2003, “
Benchmarking of Computational Fluid Dynamics for Ship Flows: The Gothenburg 2000 Workshop
,”
J. Ship Res.
0022-4502,
47
(
1
), pp.
63
81
.
20.
Kreyszig
,
E.
, 1993,
Advanced Engineering Mathematics
,
7th ed.
,
Wiley
,
New York
, Chaps. 23 and 24, pp.
1148
1271
.
21.
Ross
,
S. M.
, 2003, “
Peirce’s Criterion for the Elimination of Suspect Experimental Data
,”
J. Eng. Technol.
,
20
(
2
), pp.
38
41
.
22.
Eça
,
L.
, and
Hoekstra
,
M.
, 2000, “
An Evaluation of Verification Procedures for Computational Fluids Dynamics
,” Instituto Superior Tecnico Report No. D72-7.
23.
Bruneau
,
C. H.
, and
Saad
,
M.
, 2006, “
The 2D Lid-Driven Cavity Problem Revisited
,”
Comput. Fluids
0045-7930,
35
(
3
), pp.
326
348
.
24.
Botella
,
O.
, and
Peyret
,
R.
, 1998, “
Benchmark Spectral Results on the Lid-Driven Cavity Flow
,”
Comput. Fluids
0045-7930,
27
(
4
), pp.
421
433
.
25.
Hortmann
,
M.
,
Perić
,
M.
, and
Scheuerer
,
G.
, 1990, “
Multigrid Benchmark Solutions for Laminar Natural Convection Flows in Square Cavities
,” in
Benchmark Test Cases for Computational Fluid Dynamics
,
I.
Celik
and
C. J.
Freitas
, eds.,
ASME
,
New York
, pp.
1
6
.
26.
Celik
,
I.
, and
Karatekin
,
O.
, 1997, “
Numerical Experiments on Application of Richardson Extrapolation With Nonuniform Grids
,”
ASME J. Fluids Eng.
0098-2202,
119
(
3
), pp.
584
590
.
27.
Thangam
,
S.
, and
Speziale
,
C. G.
, 1992, “
Turbulent Flow Past a Backward-Facing Step: A Critical Evaluation of Two-Equation Models
,”
AIAA J.
0001-1452,
30
(
5
), pp.
1314
1320
.
28.
Avva
,
R. K.
,
Kline
,
S. J.
, and
Ferziger
,
J. H.
, 1998, “
Computation of Turbulent Flow Over a Backward-Facing Step-Zonal Approach
,”
AIAA 26th Aerospace Sciences Meeting
, Reno, NV, Jan.
29.
Pérez-Segarra
,
C. D.
,
Oliva
,
A.
, and
Cònsul
,
R.
, 1996, “
Analysis of Some Numerical Aspects in the Solution of the Navier-Stokes Equations Using Non-Orthogonal Collocated Finite-Volume Methods
,”
Proceedings of the Third ECCOMAS Computational Fluid Dynamics Conference
, Paris, France,
Wiley
,
New York
, pp.
505
511
.
30.
Demirdžić
,
I.
,
Lilek
,
Ž.
, and
Perić
,
M.
, 1992, “
Fluid Flow and Heat Transfer Test Problems for Non-Orthogonal Grids: Bench-Mark Solutions
,”
Int. J. Numer. Methods Fluids
0271-2091,
15
(
3
), pp.
329
354
.
31.
Pérez-Segarra
,
C. D.
,
Oliva
,
A.
,
Costa
,
M.
, and
Escanes
,
F.
, 1995, “
Numerical Experiments in Turbulent Natural and Mixed Convection in Internal Flows
,”
Int. J. Numer. Methods Heat Fluid Flow
0961-5539,
5
(
1
), pp.
13
33
.
32.
Pérez-Segarra
,
C. D.
,
Cadafalch
,
J.
,
Rigola
,
J.
, and
Oliva
,
A.
, 1999, “
Numerical Study of Turbulent Fluid Flow Through Valves
,”
Proceedings of the International Conference on Compressors and Their Systems
,
City University
,
London
, pp.
13
14
.
33.
Cadafalch
,
J.
,
Pérez-Segarra
,
C. D.
,
Cònsul
,
R.
, and
Oliva
,
A.
, 2002, “
Verification of Finite Volume Computations on Steady-State Fluid Flow and Heat Transfer
,”
ASME J. Fluids Eng.
0098-2202,
124
(
1
), pp.
11
21
.
34.
Sommers
,
L. M. T.
, 1994, “
Simulation of Flat Flames With Detailed and Reduced Chemical Models
,” Ph.D. thesis, Technical University of Eindhoven, Eindhoven, The Netherlands.
35.
Soria
,
M.
,
Cadafalch
,
J.
,
Cònsul
,
R.
, and
Oliva
,
A.
, 2000, “
A Parallel Algorithm for the Detailed Numerical Simulation of Reactive Flows
,”
Proceedings of the 1999 Parallel Computational Fluid Dynamics Conference
, Williamsburg, VA, pp.
389
396
.
36.
Xing
,
T.
, and
Stern
,
F.
, 2010, “
Factors of Safety for Richardson Extrapolation
,” IIHR Report No. 476.
37.
Xing
,
T.
,
Carrica
,
P.
, and
Stern
,
F.
, 2008, “
Computational Towing Tank Procedures for Single Run Curves of Resistance and Propulsion
,”
ASME J. Fluids Eng.
0098-2202,
130
(
10
), pp.
101102
.
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