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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
Building confidence to CFD results
Verification Validation
Code/Program verification
Verification of Calculation
Validation ofIdealized 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/PartialFlow Path
• Actual Hardware[Sindir et al.]
Validation ofactual
configuration
FRECON (FDM) FLUENT (FVM) FIDAP (FEM) SOLVSTR (FDM) SOLVMEF (MEF)
Ra = 1.5 · 106 Pr = 13.31
BENCHMARK DEFINITIONFOR 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:
Th = 10C Tc = 0C
VERIFICATION PROCEDURE
Reference solution
Error indicator for code comparisons
N
iii xwxf
Nf
1
2)()(1
CALCULATE: SOLUTION S , SOLUTION UNCERTAINTY USN
N
iii xwxf
Nf
1
2)()(1
INTER-CODE COMPARISONS
U,W along Y=0.5L U,W along X=0.5L U,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)
SENSITIVITY ANALYSIS
Boundary conditionsTH, TC, Text, Q1, Q2, Q3
Initial conditionsTinit. ,vinit
Material properties,,,,cp
MODEL
COMP. RESULTSINITIAL PARAMETERS
i
NiNii
i
pppFpppFDF
,...,,...,,...,,..., 11
Ni
NiNiid pppF
pppFpppFF
,...,,...,
,...,,...,,...,,...,)(
1
11
SENSITIVITY MEASURESOUTPUT
1. Fundamental parameters for validation procedure
2. Precision of measurements necessary to validate
calculations
CAVITY
CENTRAL CROS-SECTION
AL
UM
INIU
M
W
AL
L
AL
UM
INIU
M
W
AL
L
PLEXIGLASS WALL
PLEXIGLASS WALL
T7 T10
T14
T15
Th
TL TP
Tc
TE1 TE2
• Particle Image Velocimetry
• Particle Image Thermometry
• 2D Visualization
• Point temperature measurements
MEASUREMENTS TECHNIQUES
correlationF(t0)
F(t0+t)
Niiavg v
Nv
..1
1
21
..1
2
1
1
Ni
avgiN vvN
ESTIMATION OF EXP. UNCERAINTY UD
21
..1
2
11
Niavgi vv
NNs
• PIVAvg. Fields N – length of series
Std. Dev.
Std. Dev. Error
Experimental Data Uncertainty
• PIT
svsvUvUv avgavgDavgDavg 3;3;
sUD 3
Halcrest Inc. B
M100
Temp. range [C] Hue Color UD[C]
5.5 6.4 0.12 0.28 Red 1.0
6.4 6.5 0.28 0.35 Yellow 0.5
6.5 7.5 0.35 0.55 Green 1.0
7.5 9.5 0.55 0.70 Blue 1.5
EXPERIMENTAL BENCHMARKTwo Liquid Crystals cover entire color range [0 C, 10 C]
Th = 10 C Tc = 0 C
PIV
PIT
Ra = 1.5*106
Pr = 11.78
EXPERIMENTAL BENCHMARK
2D Temp. Field Temp. along Y = 0.5L Temp. along X = 0.9L
W along Y = 0.5L U along X = 0.5L W along X = 0.9L
EXPERIMENTAL UNCERTAINTY ESTIMATION
Niiavg v
Nv
..1
1
21
..1
2
11
Niavgi vv
NNs
smmyxs /18.080,0:3max
N = 40, t = 1s
Mix C
Temp. range [C] Hue Color UD[C]
0.0 3.0 0.11 0.18 Red 1.0
3.0 3.5 0.18 0.25 Yellow 0.5
3.5 3.9 0.25 0.48 Green 0.5
3.9 8.0 0.48 0.66 Blue 3.0
Halcrest Inc. B
M100
5.5 6.4 0.12 0.28 Red 1.0
6.4 6.5 0.28 0.35 Yellow 0.5
6.5 7.5 0.35 0.55 Green 1.0
7.5 9.5 0.55 0.70 Blue 1.5
• PIV
• PIT
s
• Comparison Error
• Validation metric
SDE
VALIDATION METHODOLOGY
Stern et all., Comprehensive approach to verification and validation of CFD simulations – Part 1: Methodology and proceduresJournal of Fluids Engineering – Transactions of ASME, 123 (4), pp. 793-802,2001.
5.0222SPDSNDV UUUUE
5.0222SPDSNDV UUUU
sUD 3 SSSU extSN
21
..1
2
11
Niavgi vv
NNs
0SPDU
Niiavg v
Nv
..1
1 cfext SSS 33.033.1
In our example:
for water
VALIDATION EXAMPLE
Simulation Avariable material properties of water
,,cp
Simulation Bconst. material properties of water
,,cp = const.
Simulation CAdiabatic and isothermal walls
Tem
pera
ture
fie
lds
Vel
ocity
Fie
lds
THERMAL BOUNDARY CONDITION VALIDATION
Th=
10C
Tc=
- 2C
Computational Simulation
Experiment
extwii TTQ
121 10 KWm
122 2400 KWm
123 1000 KWm
THERMAL BOUNDARY CONDITION VALIDATION
Computational Simulation
ExperimentT
h=
10C
Tc
= -
1C
extwii TTQ
121 10 KWm
122 2400 KWm
123 1000 KWm
THERMAL BOUNDARY CONDITION VALIDATION
Th=
10C
Tc=
1C
Computational Simulation
Experiment
extwii TTQ
121 10 KWm
122 2400 KWm
123 1000 KWm
THERMAL BOUNDARY CONDITION VALIDATION
Th=
10C
Tc=
2C
Computational Simulation
Experiment
extwii TTQ
121 10 KWm
122 2400 KWm
123 1000 KWm
VALIDATION – QUANTITATIVE COMPARISONS
Tem
per
atu
re p
rofi
les
Vel
oci
ty p
rofi
les
Y=0.5L X=0.5L X=0.9L
NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS
Th
= 2
7.33
C
Tc
= 6
.87 C
Th
= 2
7.21
C
Tc
= 6
.77 C
Ra Pr1 3*107 9.53
2 1.5 *108 7.01
3 1.8*108 7.01
4 4.4*108 5.41
avg
N
vI
21
..1
2
1
1
Ni
avgiN vvN
Niiavg v
Nv
..1
1
Turbulence Intensity
N = 150
t = 100 ms
t = 15 sec
Ra = 4.4x108
Ra = 1.5x108
Ra = 1.8x108
Ra = 3x107
NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS
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