Assessment of Discretization Uncertainty Estimators Based ...
Transcript of Assessment of Discretization Uncertainty Estimators Based ...
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