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

EXPERIMENTAL SET-UP

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

Ra = 3x107

Ra = 4.4x108

NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

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

Ra = 3x107

N=150 t = 100 ms

NATURAL CONVECTIVE FLOW FOR HIGH RAYLEIGH NUMBERS

Ra = 4.4x108

N=138 t = 100 ms

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