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 PROCEDURECompare profiles (not points!)

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 COMPARISONSusing selected profiles

Error U,W along Y=0.5L Error U,W along X=0.5L Error 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)

Mesh sensitivity

SENSITIVITY ANALYSISParameters and control points

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

light sheet

CAVITY DETAILSControl points for monitoring internal and external temperatures

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 (PIV)

Particle Image Thermometry (PIT)

2D VisualizationPoint temperature measurements

EXPERIMENTAL 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 BENCHMARK DEFINED Different liquid crystal tracers to cover entire color range

Th = 10 C Tc = 0 C

PIV – velocity

PIT -temperature

Ra = 1.5*106

Pr = 11.78

EXPERIMENTAL BENCHMARK DEFINEDSelected velocity and temperature profiles

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

BM

100

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

• PITtwo sets of tracers

s

• Validation 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

TUNNING NUMERICAL SOLUTIONEffect of fluid variable properties and thermal boundary conditions

Simulation AVariable liquid properties

(T),(T),cp (T)

Simulation BConst. liquid properties

,,cp = const.

Simulation CAdiabatic and isothermal walls

,,cp = const

Tem

pera

ture

fie

lds

Vel

ocity

fie

lds

THERMAL BOUNDARY CONDITION Validation of the selected numerical model for Tc=-2oC

Th=

10C

Tc=

- 2C

Computational Simulation

Experiment

extwii TTQ

121 10 KWm

122 2400 KWm

123 1000 KWm

THERMAL BOUNDARY CONDITION Validation of the selected numerical model for Tc=-1oC

Computational Simulation

ExperimentT

h=

10C

Tc

= -

1C

extwii TTQ

121 10 KWm

122 2400 KWm

123 1000 KWm

THERMAL BOUNDARY CONDITION Validation of the selected numerical model for Tc=+1oC

Th=

10C

Tc=

1C

Computational Simulation

Experiment

extwii TTQ

121 10 KWm

122 2400 KWm

123 1000 KWm

THERMAL BOUNDARY CONDITION Validation of the selected numerical model for Tc=+2oC

Th=

10C

Tc=

2C

Computational Simulation

Experiment

extwii TTQ

121 10 KWm

122 2400 KWm

123 1000 KWm

VALIDATION – QUANTITATIVE COMPARISONS WITH THE EXPERIMENTAL BENCHMARK

Tem

per

atu

re p

rofi

les

Vel

oci

ty p

rofi

les

Y=0.5L X=0.5L X=0.9L

NATURAL CONVECTION AT HIGH RAYLEIGH NUMBER

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

PIV PIT with two TLCs

Ra = 3.107

Ra = 4.4.108

NATURAL CONVECTION AT HIGH RAYLEIGH NUMBER

control points and area selectedfor velocity measurements

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

HIGH RAYLEIGH NUMBERVelocity field statistics

Ra = 3x107

N=150 t = 100 ms

HIGH RAYLEIGH NUMBERVelocity histogram and time series

Ra = 4.4x108

N=138 t = 100 ms

HIGH RAYLEIGH NUMBERVelocity histogram and time series

CONCLUSIONS

Numerical benchmark based on natural convection of freezing water defined

A sensitivity analysis proposed to evaluate effects of initial parameters and to identify fundamental (crucial) parameters => determination of measurement’s precision needed in the validation procedure.

Uncertainty of experimental data assessed

2D Temperature field, 2D Velocity field obtained for defined configuration

Validation procedure performed in order to assess modeling errors.

Experimental benchmark defined

High Rayleigh number natural convection resolved experimentally – Numerical solution … pending

Thank you for your attention!