Tomasz Michałek, Tomasz A. Kowalewski
NUMERICAL BENCHMARK BASED ON NATURAL
CONVECTION OF FREEZING WATER
Institute of Fundamental Technological Research
Polish Academy of Sciences,
Dept. of Mechanics and Physics of Fluids, Poland.
Building confidence to CFD results
Verification
Code/Program
verification
•Method of
manufactured solution
[Roache]
Verification
of Calculation
• Richardson
extrapolation (RE)
Validation
Validation of
Idealized
problems
Validation of
actual
configuration
• Unit problems
• Benchmark cases
•Analytical solutions
•Generalized RE
[Stern at all.]
•Numerical benchmarks
[Ghia, de Vahl Davis,
Le Quere,…]
• Simplified/Partial
Flow Path
• Grid Convergence
Index (GCI) [Roache]
• Actual Hardware
[Sindir et al.]
sensitivity
analysis
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:
 FRECON (FDM)
 FLUENT (FVM)
 FIDAP (FEM)
 SOLVSTR (FDM)
 SOLVMEF (MEF)
Th = 10C
Tc = 0C
Ra = 1.5 · 106 Pr = 13.31
VERIFICATION PROCEDURE
Compare profiles (not points!)
CALCULATE: SOLUTION S , SOLUTION UNCERTAINTY USN
Error indicator
for code comparisons
1 N
2
  f     f ( xi )  w( xi ) 
N i1
Reference solution
INTER-CODE COMPARISONS
using selected profiles
FRECON3V (FRE)
1 N
2
FLUENT 6.1. (FLU)
  f     f ( xi )  w( xi ) 
FIDAP 8.7.0.(FID)
N i1
SOLVSTR (STR)
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
Error U,W along Y=0.5L
Error U,W along X=0.5L
Mesh sensitivity
Error U,W along X=0.9L
SENSITIVITY ANALYSIS
Parameters and control points
COMP. RESULTS
INITIAL PARAMETERS
Boundary conditions
TH, TC, Text, Q1, Q2, Q3
Initial conditions
Tinit. ,vinit
Material properties
,,,,cp
MODEL
OUTPUT
SENSITIVITY MEASURES
1. Fundamental parameters
for validation procedure
2. Precision of measurements
necessary to validate
calculations
DF


i

 d ( F ) 
F  p1 ,..., pi   i ,..., pN   F  p1 ,..., pi ,..., pN 
i
F  p1 ,..., pi   i ,..., p N   F  p1 ,..., pi ,..., p N 
F  p1 ,..., pi ,..., p N 
EXPERIMENTAL SET-UP
light sheet
CAVITY DETAILS
Control points for monitoring internal and external temperatures
CENTRAL CROS-SECTION
TE1
TE2
T14
PLEXIGLASS WALL
ALUMINIUM
TL
Tc
ALUMINIUM
Th
WALL
T10
WALL
T7
PLEXIGLASS WALL
T15
TP
EXPERIMENTAL TECHNIQUES
Particle Image Velocimetry (PIV)
F(t0)
correlation
F(t0+t)
Particle Image Thermometry (PIT)
2D Visualization
Point temperature measurements
ESTIMATION OF EXP. UNCERAINTY UD
• PIV
1

vavg 
N
Avg. Fields
1
2

1
  2 

s  
v

i  vavg 
 N  N  1 i 1.. N

Std. Dev. Error
Experimental
Data Uncertainty
Halcrest Inc. BM100
• PIT
N – length of series
i 1.. N
  2
 1


v

i  v avg  
 N  1 i 1.. N

N
Std. Dev.

v
i
v
1
2
U D  3s
 
  U  v  3s ; v  3s

U
;
v
avg
D
avg
D
avg

avg

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
PIT temperature
Ra = 1.5*106
Pr = 11.78
PIV –
velocity
Th = 10 C
Tc = 0 C
EXPERIMENTAL BENCHMARK DEFINED
Selected velocity and temperature profiles
2D Temp. Field
Temp. along Y = 0.5L
W along Y = 0.5L
U along X = 0.5L
Temp. along X = 0.9L
W along X = 0.9L
EXPERIMENTAL UNCERTAINTY ESTIMATION
• PIV
1

vavg 
N

v
i
s
N = 40, t = 1s
i 1.. N

1
  2 

s  
v

i  vavg 
 N  N  1 i 1.. N

1
2
max 3s : 0  x, y  80  0.18 mm / s
Temp. range [C]
Mix C
• PIT
two sets of tracers
Hue
Color
UD[C]
BM100
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
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
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.
• Validation error
E  DS

UV  U  U
• Validation metric

E  UV  U  U
2
D
2
D
2
SN
2
SN
U
U

0.5
2
SPD

0.5
2
SPD
In our example:
U D  3s
1

vavg 
N

v
i
i 1.. N

1
  2 

s  
v

i  vavg 


N
N

1
i 1.. N


1
2
U SN   S   Sext  S
U SPD  0
S ext  1.33S f  0.33S c
for water
TUNNING NUMERICAL SOLUTION
Effect of fluid variable properties and thermal boundary conditions
Simulation A
Simulation B
Simulation C
Variable liquid properties
Const. liquid properties
Adiabatic and isothermal walls
Velocity fields
Temperature fields
(T),(T),cp (T)
,,cp = const.
,,cp = const
THERMAL BOUNDARY CONDITION
Validation of the selected numerical model for Tc=-2oC
Computational
Simulation
Qi   i Tw  Text 
1  10Wm 2 K 1
Experiment
Tc= - 2C
3  1000Wm 2 K 1
Th=10C
2  2400Wm 2 K 1
THERMAL BOUNDARY CONDITION
Validation of the selected numerical model for Tc=-1oC
Computational
Simulation
Qi   i Tw  Text 
1  10Wm 2 K 1
Experiment
Tc = -1C
3  1000Wm 2 K 1
Th=10C
2  2400Wm 2 K 1
THERMAL BOUNDARY CONDITION
Validation of the selected numerical model for Tc=+1oC
Computational
Simulation
Qi   i Tw  Text 
1  10Wm 2 K 1
Experiment
Tc=1C
3  1000Wm 2 K 1
Th=10C
2  2400Wm 2 K 1
THERMAL BOUNDARY CONDITION
Validation of the selected numerical model for Tc=+2oC
Computational
Simulation
Qi   i Tw  Text 
1  10Wm 2 K 1
Experiment
Tc=2C
3  1000Wm 2 K 1
Th=10C
2  2400Wm 2 K 1
Velocity profiles
Temperature profiles
VALIDATION – QUANTITATIVE COMPARISONS WITH
THE EXPERIMENTAL BENCHMARK
Y=0.5L
X=0.5L
X=0.9L
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
Tc = 6.87 C
1
Tc = 6.77 C
Pr
Th = 27.21 C
Ra
Th = 27.33 C
NATURAL CONVECTION
AT HIGH RAYLEIGH NUMBER
NATURAL CONVECTION
AT HIGH RAYLEIGH NUMBER
Ra = 3.107
control points and area selected
for velocity measurements
Ra = 4.4.108
HIGH RAYLEIGH NUMBER
Velocity field statistics
Ra = 3x107
Ra = 1.5x108
Turbulence Intensity
N
I 
vavg
N
 1
vi  vavg 2 


 N  1 i 1.. N

1

vavg 
N

v
i
i 1.. N
N = 150
t = 100 ms
t = 15 sec
1
2
Ra = 1.8x108
Ra = 4.4x108
HIGH RAYLEIGH NUMBER
Velocity histogram and time series
Ra = 3x107
N=150
t = 100 ms
HIGH RAYLEIGH NUMBER
Velocity histogram and time series
Ra = 4.4x108
N=138
t = 100 ms
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.
Experimental benchmark defined
2D Temperature field, 2D Velocity field obtained for defined configuration
Uncertainty of experimental data assessed
Validation procedure performed in order to assess modeling errors.
High Rayleigh number natural convection resolved experimentally –
Numerical solution … pending
Thank you for your attention!