Discretization error analysis with unfavorable meshes A case study

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1 Discretization error analysis with unfavorable meshes A case study Denis F. Hinz and Mario Turiso Kamstrup A/S, Denmark ASME 2016 V&V Symposium May 16 20, 2016, Las Vegas, Nevada

2 Agenda Who we are Background and motivation Case study Results with two approaches GCI Least squares Conclusions 2

3 Who we are The world s leading supplier of intelligent energy and water metering solutions 3

4 Our customer segments Water solutions Heating solutions Cooling solutions Meters AMR/AMI solutions Hosting & services Analytics Electricity solutions 4

5 Our locations 23 Offices located in 23 countries 40+ Distributors in 40+ countries 5

6 Fully automated production Flexibility Scalability Quality 6

Marina Bay Sands Hotel Singapore 1500 metering points Water, cooling & electricity meters Remote reading with M-Bus Reliability Kamstrup metering complements well with our Honeywell Systems

7 Marina Bay Sands Hotel Singapore 1500 metering points Water, cooling & electricity meters Remote reading with M-Bus Reliability Kamstrup metering complements well with our Honeywell Systems Solutions, enhances our tenant billing system package, provides instant data for energy savings analysis and simplifies the billing process. Facility people can see the whole energy consumptions of the property in a glance. Joseph P. Evidente, Sales Manager Honeywell Pte Ltd. 7

Introduction and motivation Fluid dynamics governs the foundation of

Understand, predict, and control physical phenomena Key challenges: 1.

Lack of computationally cheap models 4. Experimental difficulty 5.

8 Introduction and motivation Fluid dynamics governs the foundation of our technology! Understand, predict, and control physical phenomena Key challenges: 1. Non-linear 2. Chaotic (turbulence) 3. Lack of computationally cheap models 4. Experimental difficulty 5. Goal: Accurate and robust prediction of water and energy consumption. Detailed analysis of fluid flow has huge potential to support meter development and testing. 8

Project motivation Estimation of numerical uncertainty prerequisites One-term expansion power series δ RE φ i φ 0 = αh i p Structured meshes Geometrical similar meshes Large enough refinement ratios

9 Project motivation Estimation of numerical uncertainty prerequisites One-term expansion power series δ RE φ i φ 0 = αh i p Structured meshes Geometrical similar meshes Large enough refinement ratios Results with asymptotic behavior Very difficult in practical applications! William et al. (2002): A numerical method of formal accuracy p, does not necessarily mean that the computational result will also be of order p Source: Viola et al. (2013) Eça and Hoekstra (2014): It is virtually impossible to decide whether or not a given set of data is in the asymptotic range. In the presence of scatter, an observed order of grid convergence equal to the formal order of grid convergence may be fortuitously obtained 9

10 Project motivation Can the prerequisites for uncertainty estimation be relaxed? Realistic industrial CFD simulations Unstructured meshes Lack of geometrical similarity Minimum number of Grid levels Small grid refinement ratios Scatter in the results and oscillatory behavior Formal and observed order of convergence not equal ASME V&V Grid convergence index method (GCI) Limited guidance on prerequisites Three grid refinement levels Simulation results for five grid refinement levels Eça and Hoekstra (2009) Least-squares approach Recommend for problems with noisy p Five grid refinement levels 10

Incompressible Statistically-steady RANS Computational domain CAD model of the

11 Test case Internal turbulent flow downstream from asymmetric swirl disturbance generator Re = T = 20 C D = 15 mm Three dimensional Water, Incompressible Statistically-steady RANS Computational domain CAD model of the asymmetric swirl disturbance generator Level 3 mesh cut through the disturbance generator 11

12 Flow patterns Iso-surfaces for axial velocity w=2.5m/s. Color and scale represent v y (m/s) on the iso-surface. Level 1 Contour plots of the axial velocity w (m/s) at cross- section z/d = 12 Level 3 Level 5 12

13 Test case Simulation Meshes Variables OpenFoam, SIMPLE algorithm Reynolds-averaged Navier-Stokes approach Turbulence models: RNG k-ε, k-ω SST Discretization: Second order Gaussian linear scheme snappyhexmesh Hex-dominant unstructured mesh 5 geometrical equal grid levels Grid refinement factor 1.3 > r > 1.38 Finest grid with 20 million cells Averaged values at different cross-sections downstream Pressure Axial velocity Axial velocity over a ring-section Flow performance indicators 13

14 Test case Snappy Hex Mesh Number of cells Volume h i = (V/N) 1/3 Normalized h in =h i /h finest Coarsest, 5 0.6M 7,05E-05 0, ,15 Level 4 1.4M 7,05E-05 0, ,42 r 45 1,300 Level 3 3.3M 7,05E-05 0, ,82 r 34 1,332 Level 2 8.7M 7,05E-05 0, ,32 r 23 1,382 Finest, M 7,05E-05 0, ,00 r 12 1,315 14

15 ASME V&V-20 (2009) GCI Method For a flow variable result (φ i ) and triplets of three grid refinement levels: ε 32 = φ 3 φ 2, ε 12 = φ 2 φ 1 p = 1 ln r 21 ln q p = ln r 21 p s p s r 32 s = 1 sign(ε 32 ε 21 ) ε 32 ε 21 + q(p) A p (observed order of convergence) Fixed point iteration e a 21 = φ 1 φ 2 φ 1 21 GCI fine Fs = 1.25 = Fs e a 21 p r 21 1 Celik et al. (2008) p constrained to be positive p = 1 ln r 21 ln ε 32 ε 21 + q(p) Roache (2009) general procedure (need constant r). ε 32 p r p = r 32 1 ε r p 21 1 C B 15

value 2 Finest grid triplets do not lead to p values closer to 2 Not consistent results comparing results from

16 ASME V&V-20 (2009) GCI Method Results A B C 16 Results for k-ω SST model for averaged axial velocity. A and C lead to negative values of p due to oscillatory behavior and scatter p is not close to its theoretical value 2 Finest grid triplets do not lead to p values closer to 2 Not consistent results comparing results from different triplets GCI sensitive to p, Eça and Hoekstra (2009) p < 1 GCI becomes over-conservative p > 2.5 GCI becomes too optimistic

17 Least squares: Eça and Hoekstra (2009) Least squares fit: S φ 0, α, p = N i=1 φ i (φ 0 + αh p i ) 2 δ RE Non-linear system solved with Matlab least squares curve fitting tool Alternative expressions to avoid over-conservative and super-convergence estimates: a For 0.95 p < 2.05 U d = 1.25 δ RE + U s b For 0 < p < 0.95 U d = min 1.25 δ RE + U s, 1.25 M c For p 2.05 U d = max 1.25 δ 2 + U s, 1.25 M - Standard deviation U s = Ng i=1 φ i (φ 0 + αh p i ) 2 Ng 3 - Maximal difference between solutions M = φ max φ min 17 - One-term expansion with p = 2 δ 2 φ i φ 0 = αh i 2

18 Least squares: Eça and Hoekstra (2009) Results Least squares fit p=1.77 Least squares fit p=2 Asymmetry Factor ka z=12d h=1 φ Least squares fit p α φ δ RE U s U s / δ RE 1.30 M U s / M 0.21 Numerical uncertainty U d ± U d (% of φ 1 ) 15.95% a 18

19 Least squares: Eça and Hoekstra (2009) U s greatly increase uncertainty estimation U s one or two orders of magnitude larger than δ RE Not optimal fit, scatter Least squares fit p=1.59 Least squares fit p=2 Avg. Velocity z=12d h=1 φ 1 1,01900 Least squares fit p α φ δ RE U s U s / δ RE M U s / M 0.73 Numerical uncertainty U d ± U d (% of φ 1 ) 0.54% a 19

20 Least squares: Eça and Hoekstra (2009) Limiting δ RE 0.95 p < 2.05 avoid over-conservative and super-convergence estimations of GCI method Avg. pressure z=12d h=1 φ Least squares fit p α φ δ RE U s U s / δ RE 0.43 M U s / M 0.31 Numerical uncertainty U d ± U d (% of φ 1 ) % b Least squares fit p=0.66 Least squares fit p=2 20

21 Least squares: Eça and Hoekstra (2009) Least squares fit p=3.92 Least squares fit p=2 Profile factor k p z=12d h=1 φ Least squares fit p α φ δ RE U s U s / δ RE M U s / M 0.14 δ p= Numerical uncertainty U d ± U d (% of φ 1 ) % c 21

22 Least squares: Eça and Hoekstra (2009) Values of p are more consistent and closer to theoretical value 2 than with GCI Influence of grid quality and scatter Dependency on the apparent non-monotonic convergence of the results Other cases supporting least squares approach use 6 to 15 grid levels for a total grid refinement of 2 This case: 5 grid levels for a total grid refinement of

23 Conclusions Can the prerequisites for uncertainty estimation be relaxed? Realistic industrial CFD simulations Unstructured meshes Number of grid levels Scatter in the results and oscillatory behavior Formal and observed order of convergence not equal Quality of data influence the estimation Apparent order of convergence depends on each variable High standard deviations Unstructured grids and inconsistent refinement factors Not only discretization error Code verification (OpenFoam), round-off error and iterative error 23

24 Think forward Denis F. Hinz, Ph.D. Head of Flow Laboratory

25 Appendix A. Extra slides 25

26 Appendix A. Least squares k-ω SST Avg. Pressure z=12d Row results1 phi (1) 0,01700 p 0,66229 alpha 0,03020 phi_0 0,00000 Delta 0,03020 SigmaDelta 0,01312 SigmaDelta / Delta 0,43 Max_dif 0,04200 Scatter, too coarse 31% Numerical uncertainty 0,05087 U (%) 299,21% 26

27 Appendix A. Least squares k-ω SST Avg. Velocity z=12d Row results1 phi (1) 1,01900 p 1,59816 alpha -0,00033 phi_0 1,01577 Delta 0,00033 SigmaDelta 0,00508 SigmaDelta / Delta 15,25 Max_dif 0,00700 Scatter, too coarse 73% Numerical uncertainty 0,00549 U (%) 0,54% 27

28 Appendix A. Least squares k-ω SST Avg. Ring Velocity z=12d Row results1 phi (1) 1,05500 p 4,62334 alpha 0,00002 phi_0 1,05046 Delta 0,00002 SigmaDelta 0,00458 SigmaDelta / Delta 271,56 Max_dif 0,00800 Scatter, too coarse 57% Numerical uncertainty 0,02400 U (%) 2,27% 28

29 Appendix A. Least squares k-ω SST Profile Factor kp z=12d Row results1 phi (1) 0,26100 p 3,92759 alpha 0,00379 phi_0 0,27287 Delta 0,00379 SigmaDelta 0,04928 SigmaDelta / Delta 13,01 Max_dif 0,36100 Scatter, too coarse 14% Numerical uncertainty 0,45125 U (%) 172,89% 29

30 Appendix A. Least squares k-ω SST Asymmetry Factor ka z=12d Row results1 phi (1) 0,94700 p 1,77351 alpha 0,05934 phi_0 0,84627 Delta 0,05934 SigmaDelta 0,07686 SigmaDelta / Delta 1,30 Max_dif 0,37200 Scatter, too coarse 21% Numerical uncertainty 0,15103 U (%) 15,95% 30

31 Appendix A. Least squares k-ω SST Turbulence Factor ktu z=12d Row results1 phi (1) 1,48800 p 2,70819 alpha -0,00223 phi_0 1,51594 Delta 0,00223 SigmaDelta 0,08640 SigmaDelta / Delta 38,71 Max_dif 0,06800 Scatter, too coarse 127% Numerical uncertainty 0,20400 U (%) 13,71% 31

32 Appendix A. Least squares k-ω SST Swirl Angle z=12d Row results1 phi (1) 15,53100 p 0,00395 alpha -540,66059 phi_0 556,04416 Delta 540,66059 SigmaDelta 1713,65623 SigmaDelta / Delta 3,17 Max_dif 2,44500 Scatter, too coarse 70088% Numerical uncertainty 7,33500 U (%) 47,23% 32

33 Appendix B. Least squares RNG k-ε Avg. Pressure z=12d Row results1 phi (1) 0,09500 p 0,68811 alpha -0,03188 phi_0 0,12928 Delta 0,03188 SigmaDelta 0,16272 SigmaDelta / Delta 5,10 phi Max_dif 0,03900 Scatter, too coarse 417% Numerical uncertainty 0,11700 U (%) 123,16% 33

34 Appendix B. Least squares RNG k-ε Avg. Velocity z=12d Row results1 phi (1) 1,05600 p 0,13238 alpha -0,25426 phi_0 1,30641 Delta 0,25426 SigmaDelta 0,87128 SigmaDelta / Delta 3,43 phi Max_dif 0,04100 Scatter, too coarse 2125% Numerical uncertainty 0,12300 U (%) 11,65% 34

35 Appendix B. Least squares RNG k-ε Avg. Ring Velocity z=12d Row results1 phi (1) 1,13400 p 0,07387 alpha -0,86635 phi_0 1,99017 Delta 0,86635 SigmaDelta 2,86251 SigmaDelta / Delta 3,30 phi Max_dif 0,07700 Scatter, too coarse 3718% Numerical uncertainty 0,23100 U (%) 20,37% 35

36 Appendix B. Least squares RNG k-ε Profile Factor kp z=12d Row results1 phi (1) 0,65800 p 0,01400 alpha -11,16669 phi_0 11,76489 Delta 11,16669 SigmaDelta 35,60198 SigmaDelta / Delta 3,19 phi Max_dif 0,26600 Scatter, too coarse 13384% Numerical uncertainty 0,79800 U (%) 121,28% 36

37 Appendix B. Least squares RNG k-ε Asymmetry Factor ka z=12d Row results1 phi (1) 0,95000 p 0,18581 alpha 1,13619 phi_0 0,00000 Delta 1,13619 SigmaDelta 0,20460 SigmaDelta / Delta 0,18 phi Max_dif 0,43300 Scatter, too coarse 47% Numerical uncertainty 0,54125 U (%) 56,97% 37

38 Appendix B. Least squares RNG k-ε Turbulence Factor ktu z=12d Row results1 phi (1) 1,72400 p 0,02791 alpha 1,77717 phi_0 0,00206 Delta 1,77717 SigmaDelta 0,07713 SigmaDelta / Delta 0,04 phi Max_dif 0,14900 Scatter, too coarse 52% Numerical uncertainty 0,18625 U (%) 10,80% 38

39 Appendix B. Least squares RNG k-ε Swirl Angle z=12d Row results1 phi (1) 10,28000 p 0,19498 alpha 10,74019 phi_0 0,00000 Delta 10,74019 SigmaDelta 0,92435 SigmaDelta / Delta 0,09 phi Max_dif 2,82100 Scatter, too coarse 33% Numerical uncertainty 3,52625 U (%) 34,30% 39

Richardson Extrapolation-based Discretization Uncertainty Estimation for Computational Fluid Dynamics

Accepted in ASME Journal of Fluids Engineering, 2014 Richardson Extrapolation-based Discretization Uncertainty Estimation for Computational Fluid Dynamics Tyrone S. Phillips Graduate Research Assistant