Assessment of DiscretizationUncertainty Estimators

Based on Grid Refinement Studies

L.Eça (IST/MARIN Academy), G.Vaz (MARIN), M.Hoekstra (Consultant)

S.Doebling (LANL), R.L.Singleton, Jr (LANL), G.Srinivasan (LANL)

G.Weirs (Sandia National Laboratories),

T.Phillips (University of British Columbia), C.J.Roy (Virginia Tech)

Contents

• Motivation

• Test Cases

• Overview of Methods

• Results

• Final remarks

Motivation

• Check the performance of procedures basedon grid refinement studies for the determinationof the numerical (discretization) uncertainty inflows with practical interest, i.e. high Reynoldsnumber turbulent flows:

─ Strict geometrically similarity of the grids.

─ Selected flow quantites include functional (force coefficients) and local quantities at theboundaries and in the interior of the domain.

Test Cases• Steady, two-dimensional flows of an incompressible,

one-phase, Newtonian fluid:

- Flow over a flat plate for Reynolds numbers ofRe=107(I), Re=108 (II) and Re=109 (III).

Test Cases• Steady, two-dimensional flows of an incompressible,

one-phase, Newtonian fluid:

- Flow around a NACA 0012 foil at angles of attack of α=0o (IV), α=4º(V) and α=10º(VI) with Re=6×106.

Test Cases

• Time-averaged Navier-Stokes equations (RANS)supplemented with three eddy-viscosity models:

a) Spalart-Allmaras one-equation model (SPAL);

b) SST k-ω two-equation model (SST);

c) � − ��

�two−equation model (KSKL).

• Results have negligible influence of round-off errorsand (almost) negligible influence of iterative errors.

• Extra refined grids to produce an “exact solution”.Finest grids have hi/h1=0.364 for flat plate casesand hi/h1=0.333 for NACA 0012 airfoil. hi/h1=1 isthe finest grid used for the error estimation.

Test Cases• Selected quantities of interest

Functional (integral) quantities:- Friction resistance/drag coefficient (plate, airfoil).- Pressure drag/coefficient (airfoil).- Friction and pressure lift coefficients (airfoil).

Surface quantities:- Skin friction (plate and airfoil) and pressure (airfoil)coefficients.

Interior quantities:- Horizontal and vertical velocity components andeddy-viscosity.

Test Cases

ri

CF×

10

3

0 1 2 3 4 5 6 7 82.6

2.65

2.7

2.75

2.8

2.85

2.9 Flat Plate, Re=107, SST (Case Ib)

ri

CF×

10

3

0 1 2 3 4 5 6 7 82.6

2.65

2.7

2.75

2.8

2.85

2.9

"Exact"

p= 1.3

Test Cases

ri

νt/ν

0 1 2 3 4 5 6 7 815

16

17

18

19

20

ri

νt/ν

0 1 2 3 4 5 6 7 815

16

17

18

19

20

"Exact"

p= 1.0

Flat Plate, Re=108, KSKL, Case IIc

x/L=1.0, y/L=0.000018

Test Cases

ri

Cf×

10

3

0 1 2 3 4 5 6 7 81.25

1.3

1.35

1.4 Flat Plate, Re=109, x/c=0.95

SST, Case IIIb

ri

Cf×

10

3

0 1 2 3 4 5 6 7 81.25

1.3

1.35

1.4

"Exact"

p= 1.8

Test Cases

ri

-Cp

0 0.5 1 1.5 2 2.5 3 3.5 40.0663

0.06632

0.06634

0.06636

0.06638

0.0664

NACA 0012, α=0o

x*/c=0.775, Lower surfaceSPAL, Case IVa

ri

-Cp

0 0.5 1 1.5 2 2.5 3 3.5 40.0663

0.06632

0.06634

0.06636

0.06638

0.0664

"Exact"

p= 1.0

Test Cases

ri

Vy/V

∞

0 0.5 1 1.5 2 2.5 3 3.5 4-0.084

-0.0835

-0.083

-0.0825

-0.082

ri

Vy/V

∞

0 0.5 1 1.5 2 2.5 3 3.5 4-0.084

-0.0835

-0.083

-0.0825

-0.082

"Exact"

p= 2.0

NACA 0012, α=4o, SPAL, Case Va

x/L=0.328, y/L=0.040

Test Cases

ri

Vx/V

∞

0 0.5 1 1.5 2 2.5 3 3.5 40

0.02

0.04

0.06

0.08

0.1

ri

Vx/V

∞

0 0.5 1 1.5 2 2.5 3 3.5 40

0.02

0.04

0.06

0.08

0.1

"Exact"

p= 1.0

NACA 0012, α=10o, SST, Case VIb

x/L=0.994, y/L=-0.175

Overview of Methods• FS Xing T. and Stern F., “Factors of Safety for

(3) Richardson Extrapolation” J. Fluids Eng.,132,2010.

• FSa Xing T. and Stern F., “Closure to Discussion of

(3) Factors of Safety for Richardson Extrapolation” J. Fluids Eng., 133, 2011

• GCI1 Roache P.J., “Verification and Validation in

(3) Computational Science and Engineering”,Hermosa, 1998.

• GCI2 Standard for Verification and Validation in

(3) Computational Fluid Dynamics and Heat Transfer

ASME 2009/2016

Overview of Methods• GDU Phillips T. S., Roy C. J., “A New

(3) Extrapolation-Based Uncertainty Estimator

for Computational Fluid Dynamics" Journal of V&V and UQ, Vol 1, 2017.

• LSGCI Eça L., Hoekstra M., “A procedure for the

(5) estimation of the numerical uncertainty of

CFD calculations based on grid refinement

studies” JCP 262, 2014.

• LSGCIr Modified version of the previous method(5) including “robust fits”

• RMR5 Rider, W. J., Witkowski W., Kamm J. R.RMR3 and Wildey T., “Robust Verification Analysis”

JCP 307, 2016.

Overview of Methods• For most methods, discretization error δφ is defined

as a function of the grid refinement ratio by

δφ = �

�

.

• Observed order of grid convergence p from a gridtriplet is determined from

�����

�����=

�

�

����

���

����

���

.

• FS, FSa, GCI1 and GCI2 do not provide an uncertaintyestimate when � ≤ 0 (apparent divergence).

Overview of Methods• Uncertainty estimates are obtained from

�� = �� �� .

• For RMR5 and RMR3 ��=1.

• Goal of the estimated uncertainties is to satisfy

� − �� ≤ �exact ≤ � + ��

95 out of 100 times it is tested.

Results• Uncertainty estimates evaluated using:

Fno is the percentage of cases that lead to apparentdivergence based on a grid triplet;

R is the ratio between estimated uncertaintiesand “exact errors”

R =#$

��"�exact";

FR<1 is the percentage of cases that exhibit an estimated uncertainty smaller than the “exact error”;

Rmed is the median of the R distributions.

Results• Uncertainty estimates performed for each of the

required quantities of interests were performed withdata from grid refinement levels.

Test Cases SPAL SST KSKL Flat Plate Ia Ib Ic Re=107

IIa IIb IIc Re=108

IIIa IIIb IIIc Re=109

ri FS, Fsa, GCI1, GCI2, GDUand RMR3

LSGCI, LSGCIr and RMR5

1 r1=1., r2=1.455 and r3=2. r1=1. , r2=1.231, r3=1.455, r4=1.6, and r5=2.

2 r1=2., r2=2.909 and r3=4. r1=1. , r2=2.462, r3=2.909, r4=3.2, and r5=4.

4 r1=4., r2=5.818 and r3=8. r1=1. , r2=4.923, r3=5.818, r4=6.4, and r5=8.

Results• Uncertainty estimates performed for each of the

required quantities of interests were performed withdata from grid refinement levels.

Test Cases SPAL SST KSKL NACA 0012 IVa IVb IVc α=0º

Va Vb Vc α=4º Via VIb VIc α=10º

ri FS, Fsa, GCI1, GCI2, GDUand RMR3

LSGCI, LSGCIr and RMR5

1 r1=1., r2=1.333 and r3=2. r1=1. , r2=1.143, r3=1.333, r4=1.6, and r5=2.

2 r1=2., r2=2.667 and r3=4. r1=1. , r2=2.286, r3=2.667, r4=3.2, and r5=4.

Results• Percentage of triplets that exhibit � ≤ 0

(apparent divergence) for the selected grid triplets.

Fno ri=1 r

i=2 r

i=4 All

I,II,IIIFlat Plate

16.2 20.3 25.4 20.6

IV, V,VINACA 0012

23.1 22.4 --- 22.8

Results• Flate plate, Cases I, II, III

FR

<1

0

10

20

30

40

50

60

70

80

90

100

ri=h

i/h

1=2r

i=h

i/h

1=1 All casesr

i=h

i/h

1=4

All quantities SPAL

FR

<1

0

10

20

30

40

50

60

70

80

90

100FS

FS1

GCI1

GCI2

FR

<1

0

10

20

30

40

50

60

70

80

90

100GDU

LSGCI

LSGCIr

RMR5

RMR3

Results• Flate plate, Cases I, II, III

FR

<1

0

10

20

30

40

50

60

70

80

90

100

ri=h

i/h

1=2r

i=h

i/h

1=1 All casesr

i=h

i/h

1=4

All quantities SST

FR

<1

0

10

20

30

40

50

60

70

80

90

100FS

FS1

GCI1

GCI2

FR

<1

0

10

20

30

40

50

60

70

80

90

100GDU

LSGCI

LSGCIr

RMR5

RMR3

Results• Flate plate, Cases I, II, III

FR

<1

0

10

20

30

40

50

60

70

80

90

100

ri=h

i/h

1=2r

i=h

i/h

1=1 All casesr

i=h

i/h

1=4

All quantities KSKL

FR

<1

0

10

20

30

40

50

60

70

80

90

100FS

FS1

GCI1

GCI2

FR

<1

0

10

20

30

40

50

60

70

80

90

100GDU

LSGCI

LSGCIr

RMR5

RMR3

Results• Flate plate, Cases I, II, III

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9R

All casesri=h

i/h

1=4r

i=h

i/h

1=2

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

ri=h

i/h

1=1

R RR

FS1

FS

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9R

All casesri=h

i/h

1=4r

i=h

i/h

1=2

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

ri=h

i/h

1=1

R RR

LSGCIr

LSGCI

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9R

All casesri=h

i/h

1=4r

i=h

i/h

1=2

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

ri=h

i/h

1=1

R RR

GCI2

GCI1

GDU

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

1 2 3 4 5 6 7 8 9R

All casesri=h

i/h

1=4r

i=h

i/h

1=2

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

ri=h

i/h

1=1

R RR

RMR5

RMR3

Results• NACA 0012, Cases IV, V, VI

FR

<1

0

10

20

30

40

50

60

70

80

90

100All quantities SPAL

All casesri=h

i/h

1=2r

i=h

i/h

1=1

FR

<1

0

10

20

30

40

50

60

70

80

90

100FS

FS1

GCI1

GCI2

FR

<1

0

10

20

30

40

50

60

70

80

90

100GDU

LSGCI

LSGCIr

RMR5

RMR3

FR

<1

0

10

20

30

40

50

60

70

80

90

100All quantities SST

All casesri=h

i/h

1=2r

i=h

i/h

1=1

FR

<1

0

10

20

30

40

50

60

70

80

90

100FS

FS1

GCI1

GCI2

FR

<1

0

10

20

30

40

50

60

70

80

90

100GDU

LSGCI

LSGCIr

RMR5

RMR3

Results• NACA 0012, Cases IV, V, VI

FR

<1

0

10

20

30

40

50

60

70

80

90

100All quantities KSKL

All casesri=h

i/h

1=2r

i=h

i/h

1=1

FR

<1

0

10

20

30

40

50

60

70

80

90

100FS

FS1

GCI1

GCI2

FR

<1

0

10

20

30

40

50

60

70

80

90

100GDU

LSGCI

LSGCIr

RMR5

RMR3

Results• NACA 0012, Cases IV, V, VI

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

R

All casesri=h

i/h

1=2

1 2 3 4 5 6 7 8 9

ri=h

i/h

1=1

R R

FS1

FS

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

RR R

LSGCIr

LSGCI

All casesri=h

i/h

1=2r

i=h

i/h

1=1

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

RR R

GCI2

GCI1

GDU

All casesri=h

i/h

1=2r

i=h

i/h

1=1

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

F(R

)%

0

10

20

30

40

50

60

70

80

90

100

RR R

RMR5

RMR3

All casesri=h

i/h

1=2r

i=h

i/h

1=1

1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

Results• NACA 0012, Cases IV, V, VI

Final Remarks• Several methods for the estimation of the numerical

(discretization) uncertainty based on grid refinementmethods have been tested.

• Most of the estimates performed for the selected flow quantities were based on data outside the “asymptotic range” including several cases that donot exhibit monotonic convergence.

• There is no technical reason to present numericalsimulations of practical calculations without the indication of the numerical uncertainty.

• All data available athttp://web.tecnico.ulisboa.pt/ist12278/Discretization/Workshop_discretization_2017.htm