1. Tomasz Michałek, Tomasz A. Kowalewski Institute of Fundamental Technological Research Polish Academy of Sciences, Dept. of Mechanics and Physics of Fluids, Poland. NUMERICAL BENCHMARK BASED ON NATURAL CONVECTION OF FREEZING WATER

2. Building confidence to CFD results VerificationValidation Code/Program verification Verification of Calculation Validation of Idealized problems Method of manufactured solution [Roache] Analytical solutions Numerical benchmarks [Ghia, de Vahl Davis, Le Quere,…] Richardson extrapolation (RE) Generalized RE [Stern at all.] Grid Convergence Index (GCI) [Roache] sensitivity analysis Unit problems Benchmark cases Simplified/Partial Flow Path Actual Hardware [Sindir et al.] Validation of actual configuration

3.  FRECON (FDM)  FLUENT (FVM)  FIDAP (FEM)  SOLVSTR (FDM)  SOLVMEF (MEF) Ra = 1.5 · 10 6 Pr = 13.31 BENCHMARK DEFINITION FOR THERMAL AND VISCOUS FLOWS 2D viscous, incompressible flow driven by natural convection Navier – Stokes equations with non-linear buoyancy term (water) coupled with heat transfer Temperature gradient ΔT = 10ºC Verified programs: T h = 10  CT c = 0  C

4. VERIFICATION PROCEDURE Reference solution Error indicator for code comparisons CALCULATE: SOLUTION S, SOLUTION UNCERTAINTY U SN

5. INTER-CODE COMPARISONS U,W along Y=0.5LU,W along X=0.5LU,W along X=0.9L Details of the reference solutions w(x) Michalek T., Kowalewski T.A., Sarler B. ”Natural Convection for Anomalous Density Variation of Water: Numerical Benchmark” Progress in Computational Fluid Dynamics, 5 (3-5),pp 158-170,2005 FRECON3V (FRE) FLUENT 6.1. (FLU) FIDAP 8.7.0.(FID) SOLVSTR (STR)

6. SENSITIVITY ANALYSIS Boundary conditions T H, T C, T ext, Q 1, Q 2, Q 3 Initial conditions T init., v init Material properties , , , ,c p MODEL COMP. RESULTS INITIAL PARAMETERS SENSITIVITY MEASURES OUTPUT 1. Fundamental parameters for validation procedure 2. Precision of measurements necessary to validate calculations

7. EXPERIMENTAL SET-UP

8. CAVITY CENTRAL CROS-SECTION ALUMINIUM WALL PLEXIGLASS WALL T7T7 T 10 T 14 T 15 ThTh TLTL TPTP TcTc T E1 T E2

9. Particle Image Velocimetry Particle Image Thermometry 2D Visualization Point temperature measurements MEASUREMENTS TECHNIQUES correlation F(t 0 ) F(t 0 +  t )

10. ESTIMATION OF EXP. UNCERAINTY U D PIV Avg. FieldsN – length of series Std. Dev. Std. Dev. Error Experimental Data Uncertainty PIT Halcrest Inc. BM100 Temp. range [  C] HueColor UD[C]UD[C] 5.56.40.120.28Red1.0 6.46.50.280.35Yellow0.5 6.57.50.350.55Green1.0 7.59.50.550.70Blue1.5

11. EXPERIMENTAL BENCHMARK Two Liquid Crystals cover entire color range [0  C, 10  C] T h = 10  C T c = 0  C PIV PIT Ra = 1.5*10 6 Pr = 11.78

12. EXPERIMENTAL BENCHMARK 2D Temp. Field Temp. along Y = 0.5L Temp. along X = 0.9L W along Y = 0.5LU along X = 0.5LW along X = 0.9L

13. EXPERIMENTAL UNCERTAINTY ESTIMATION N = 40,  t = 1s Mix C Temp. range [  C] HueColor UD[C]UD[C] 0.03.00.110.18Red1.0 3.03.50.180.25Yellow0.5 3.53.90.250.48Green0.5 3.98.00.480.66Blue3.0 Halcrest Inc. BM100 5.56.40.120.28Red1.0 6.46.50.280.35Yellow0.5 6.57.50.350.55Green1.0 7.59.50.550.70Blue1.5 PIV PIT

14. Comparison Error Validation metric VALIDATION METHODOLOGY Stern et all., Comprehensive approach to verification and validation of CFD simulations – Part 1: Methodology and procedures Journal of Fluids Engineering – Transactions of ASME, 123 (4), pp. 793-802,2001. In our example: for water

15. VALIDATION EXAMPLE Simulation A variable material properties of water , ,c p Simulation B const. material properties of water , ,c p = const. Simulation C Adiabatic and isothermal walls Temperature fields Velocity Fields

16. THERMAL BOUNDARY CONDITION VALIDATION T h =10  C T c = - 2  C Computational Simulation Experiment

17. THERMAL BOUNDARY CONDITION VALIDATION Computational Simulation Experiment T h =10  C T c = -1  C

18. THERMAL BOUNDARY CONDITION VALIDATION T h =10  C T c =1  C Computational Simulation Experiment

19. THERMAL BOUNDARY CONDITION VALIDATION T h =10  C T c =2  C Computational Simulation Experiment

20. VALIDATION – QUANTITATIVE COMPARISONS Temperature profiles Velocity profiles Y=0.5LX=0.5LX=0.9L

21. VALIDATION – QUANTITATIVE COMPARISONS ExperimentComp. Sim. AComp. Sim. B YDUDUD SU SN |E||E|UVUV S |E||E|Uv 0.84.501.04.610.020.111.0004.640.020.141.000 14.05.500.55.010.010.490.5004.840.010.660.500 36.26.400.56.120.010.280.5006.020.010.380.500 44.46.450.56.570.010.120.5006.460.01 0.500 49.47.001.06.900.010.101.0006.780.010.221.000 56.37.500.57.400.010.100.5007.270.010.230.500 63.38.001.07.950.020.051.0007.850.020.151.000 69.89.500.58.650.020.850.5008.600.020.900.500 ExperimentComp. Sim. AComp. Sim. B YDUDUD SU SN |E||E|UVUV S |E||E|Uv 4.93.951.03.630.10.321.0053.690.10.261.005 18.33.900.52.640.11.260.5102.690.11.210.510 37.22.001.01.650.20.351.0201.840.20.161.020 39.73.300.51.810.21.490.5391.590.21.710.539 41.63.900.52.970.20.930.5391.520.22.380.539 42.64.000.53.890.20.110.5391.650.22.350.539 45.76.500.56.390.20.110.5393.640.22.860.539 48.37.001.07.080.10.081.0055.650.11.351.005 58.17.500.57.650.10.150.5107.590.10.090.510 64.18.001.08.140.10.141.0058.060.20.061.020 71.79.500.58.900.10.600.5108.860.10.640.510 Validation error was assessed for both simulations Assessed discrepancy are solely due to modeling errors Comp. Sim. A. turned out to be closer to experimental results than comp. Sim. B according to applied validation technique

22. NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS T h = 27.33  C T c = 6.87  C T h = 27.21  C T c = 6.77  C RaPr 13*10 7 9.53 21.5 *10 8 7.01 31.8*10 8 7.01 44.4*10 8 5.41

23. Ra = 3x10 7 Ra = 4.4x10 8 NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

24. Ra = 4.4x10 8 Ra = 1.5x10 8 Ra = 1.8x10 8 Ra = 3x10 7 Avg. Horizontal Velocity N = 150  t = 100 ms t = 15 sec NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

25. Avg. Vertical Velocity N = 150  t = 100 ms t = 15 sec Ra = 4.4x10 8 Ra = 1.5x10 8 Ra = 1.8x10 8 Ra = 3x10 7 NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

26. Skewness N = 150  t = 100 ms t = 15 sec Ra = 4.4x10 8 Ra = 1.5x10 8 Ra = 1.8x10 8 Ra = 3x10 7 NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

27. Kurtozis N = 150  t = 100 ms t = 15 sec Ra = 4.4x10 8 Ra = 1.5x10 8 Ra = 1.8x10 8 Ra = 3x10 7 NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

28. Turbulence Intensity N = 150  t = 100 ms t = 15 sec Ra = 4.4x10 8 Ra = 1.5x10 8 Ra = 1.8x10 8 Ra = 3x10 7 NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

29. Ra = 3x10 7 N=150  t = 100 ms NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

30. Ra = 1.5x10 8 N=120  t = 100 ms NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

31. Ra = 1.8x10 8 N=134  t = 100 ms NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

32. Ra = 4.4x10 8 N=138  t = 100 ms NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

33. CONCLUSIONS Numerical benchmark based on natural convection of freezing water was defined A method based on sensitivity analysis for the sake of initial parameters was devised for identification of fundamental (crucial) parameters and determination of necessary measurement’s precision needed in validation procedure. Uncertainty of experimental data were assessed 2D Temperature field, 2D Velocity field, was obtained for defined configuration Validation procedure for computational calculations was performed in order to quantitatively assess assumed modeling errors. Experimental benchmark was defined for proposed configuration