This document summarizes Roberto Cosentino's MSc thesis on uncertainty quantification of turbulent statistics from hot-wire measurements in a high pressure turbine stage. The objectives were to characterize inlet turbulence and quantify uncertainty of turbulent statistics. Hot-wire anemometry was used to measure velocity and turbulence in a high pressure turbine test facility. A moving block bootstrap method was applied to compute turbulent statistics and associated uncertainties without assumptions on the probability distributions. The analysis found the Gaussian assumption underestimates uncertainty when deviations from normality are higher.

1. Firma convenzione
Politecnico di Milano e The von Karman
Institute for Fluid Dynamics
Uncertainty Quantification of Turbulent
Statistics from Hot-Wire Measurements
Applied in a High Pressure Turbine Stage
21/12/2016
MSc. Roberto Cosentino
Relatore: Prof. P. Gaetani
Supervisors: Dr. F. Fontaneto
Dr. S. Lavagnoli

2. Roberto Cosentino, MSc. Energy Department.
Introduction
Motivations:
• Improvements in turbo-gas efficiency pass through new
concept of design and increase in inlet temperature
• The computational design process for highly performing HP
stage needs reliable boundary conditions
• Turbulence boundary conditions affect the CFD results in
flow-dynamics behavior
Objectives:
• Inlet turbulence characterization of the HP turbine stage
• Uncertainty quantification of turbulent statistics
2

3. Roberto Cosentino, MSc. Energy Department.
Outline
1. Facility Description
2. Measurement Technique
3. Results
4. Uncertainty Analysis
5. Conclusion
3

4. Roberto Cosentino, MSc. Energy Department.
CT-3 Facility – Isentropic Light Piston Compression Tube
A
A ® Tfinal = Tambient
pfinal
pinitial
æ
èç
ö
ø÷
g -1
g
4
M1 0.1
p01/p3 2.2
Twall/T01 0.68
Rpm/√T01 281.3
T01,mid 440 K

5. Roberto Cosentino, MSc. Energy Department.
CT-3 Facility – Test Section
1 2 3
5
Nb,stator 38
Nb,rotor 48
Hb [mm] 74.7

6. Roberto Cosentino, MSc. Energy Department.
CT-3 Facility – Test Evolution
6

7. Roberto Cosentino, MSc. Energy Department.
Measurement Technique – Hot-Wire Anemometry
7
• High frequency response up to 30 kHz
• High spatial resolution (l=1mm; d=9μm)
• Incompressible flow (M1 ≤ 0.1)
• Wire Temperature ≥ 550 K (Tw/T0 ≥ 1.25)
• Nickel-Platinum alloy
e'
E
= f
v'
V
;
T0 '
T0
;
r'
r
æ
èç
ö
ø÷
Hot-Wire Thermocouple

8. Roberto Cosentino, MSc. Energy Department.
Measurement Technique – Calibration
8
Eb = f V,T0( ) Nu = f Re( )

9. Roberto Cosentino, MSc. Energy Department.
Measurement Technique – Data Reduction
9

10. Roberto Cosentino, MSc. Energy Department.
Results – Mean Velocity
10
N.B. Each point is related to a different test

11. Roberto Cosentino, MSc. Energy Department.
Results – Turbulence Intensity
11
Tu =
urms
'
U

12. Roberto Cosentino, MSc. Energy Department.
Results – Integral Length Scale
12
T =
E f( )
4u'2
é
ë
ê
ù
û
ú
f ®0
*
L = TU
Taylor Frozen
Hypothesis
* P.E. Roach (1986)
BPF

13. Roberto Cosentino, MSc. Energy Department.
Results – Probability Density Function
13
• Distribution deviates from normality with a positive skewness
• The higher is the turbulence level and the lower is the integral
length scale, the lower is the skewness (tending to normality)

14. Roberto Cosentino, MSc. Energy Department.
Uncertainty Analysis – Bootstrap Principle (MBB)
14
• Monte-Carlo style
procedure for resampling
(each sample with mass
1/N)
• Asymptotically
consistent for large
number of samples and
replications B
• No need for assumption
on the probability density
function (PDF)
Moving Block Bootstrap (MBB)
ensures that correlation between
narrow samples is not destroyed by
drawing N-b+1 blocks (b block length)

15. Roberto Cosentino, MSc. Energy Department.
Uncertainty Analysis – MBB Results
15
q*
=
qi
*
i=1
B
å
B
sq
*
=
qi
*
-q*
( )
2
i=1
B
å
B -1
é
ë
ê
ê
ê
ù
û
ú
ú
ú
1/2
When B grows large enough the probability density function tends
asymptotically to normality and the mean and the standard deviation
can be computed, as well as the 95% confidence interval

16. Roberto Cosentino, MSc. Energy Department.
Uncertainty Analysis – MBB Comparison
16
s
u'2
=
u'2
2Neff
sU
=
u'2
Neff
sTu
= s
u'2
dTu
d u'2
æ
è
ç
ö
ø
÷
2
+ sU
dTu
dU
æ
èç
ö
ø÷
2
Normal distribution
assumption:

17. Roberto Cosentino, MSc. Energy Department.
Conclusion
• The measurement technique with the use of a
dimensionless methodology for calibration allows to
reduce data from a strongly non-isothermal flow
• The MBB technique allows to compute turbulent statistics
without assumptions on the real distribution and it confers
turbulent statistics with a good estimate of uncertainty
maintaining the deterministic feature of turbulence
• The Gaussian assumption under-estimates the uncertainty
when deviation from normality is higher
17

18. Firma convenzione
Politecnico di Milano e The von Karman
Institute for Fluid Dynamics
Thank you
for your attention
21/12/2016
MSc. Roberto Cosentino
Relatore: Prof. P. Gaetani
Supervisors: Dr. F. Fontaneto
Dr. S. Lavagnoli

19. Firma convenzione
Politecnico di Milano e The von Karman
Institute for Fluid Dynamics
Thank you
for your attention
21/12/2016
MSc. Roberto Cosentino
Relatore: Prof. P. Gaetani
Supervisors: Dr. F. Fontaneto
Dr. S. Lavagnoli

20. Firma convenzione
Politecnico di Milano e The von Karman
Institute for Fluid Dynamics
Thank you
for your attention
21/12/2016
MSc. Roberto Cosentino
Relatore: Prof. P. Gaetani
Supervisors: Dr. F. Fontaneto
Dr. S. Lavagnoli

21. Roberto Cosentino, MSc. Energy Department.
Hot-wire Angular Effect
I
Veff = V cos2
a +b2
sin2
aéë ùû
1/2
• For 2dw/lw ≥ 200, the
term b can be neglected
with small error
(Comte-Bellot, 2013)
• In this case dw = 9 μm
and lw = 1 mm for a
ratio 2dw/lw ≈ 222

22. Roberto Cosentino, MSc. Energy Department.
Calibration Methodology
II
M =
P0
P
æ
è
ç
ö
ø
÷
g-1
g
-1
é
ë
ê
ê
ù
û
ú
ú
2
g -1
ì
í
ï
îï
ü
ý
ï
þï
0.5
Ts = T0
Ps
P0
æ
è
ç
ö
ø
÷
g-1
g
r =
Ps
RgasTs
U = M gRgasTs
m =1.716´10-5 273.15+110.4
T0 +110.4
æ
è
ç
ö
ø
÷
T0
273.15
æ
è
ç
ö
ø
÷
3
2
Re =
rUdw
m
Q =
Eb
2
Rw
Rtot
2
k = kref
T0
Tref
æ
è
çç
ö
ø
÷÷
0.7
Nu =
Eb
2
k Tw -hT0( )
Rw
plaRtot
2
æ
è
ç
ö
ø
÷

23. Roberto Cosentino, MSc. Energy Department.
Static Pressure Iteration
III

24. Roberto Cosentino, MSc. Energy Department.
Results – Micro Length Scale
1
l2
=
2p2
U
2
u'2
f 2
E f( )df
0
¥
ò
IV
BPF

25. Roberto Cosentino, MSc. Energy Department.
Autocorrelation
V
Ruu t( )=
U t( )U t +t( )
U t( )2

26. Roberto Cosentino, MSc. Energy Department.
Uncertainty Sources
VI
2 Groups of Uncertainty
Sources following ASME
methodology
Systematic (b) Random (s)
• Manufacturer’s
Specifications
• Calibration Uncertainty
of instruments
• Noise
• Error propagation
• Flow unsteadiness in the
averaging process
U = si
2
i=1
ns
åæ
è
ö
ø
2
+ bi
2
i=1
nb
åæ
è
ö
ø
2

27. Roberto Cosentino, MSc. Energy Department.
Uncertainty Analysis – MBB Series and Block Length
• The higher the number of replications B, the more the statistics tend
to normality; depending on the statistics, B is different
• A number of sample of 1/10 times the total samples was found to be
the optimum block length
VII

28. Roberto Cosentino, MSc. Energy Department.
Automatic Block Length Selection
VIII
Ruu t( )=
U t( )U t +t( )
U t( )2
bopt = N1/3 2G2
D
æ
èç
ö
ø÷
1/3
D =
4g2
0( )
3
g 0( )= l k / 2m( )´ Ruu kDt( )
k=-2m
2m
å
G = l k / 2m( )´ k ´ Ruu kDt( )
k=-2m
2m
å
Find the smallest lag m for which for every k ≥ m the autocorrelation
function can be neglected
N.B. For autocorrelation function with wide
oscillation about zero, this method is not applicable
and a conservative value for bopt must be chosen