Abstract

Our investigation raises an important question that is of relevance to the wider turbomachinery community: how do we estimate the spatial average of a flow quantity given finite (and sparse) measurements? This paper seeks to advance efforts to answer this question rigorously. In this paper, we develop a regularized multivariate linear regression framework for studying engine temperature measurements. As part of this investigation, we study the temperature measurements obtained from the same axial plane across five different engines yielding a total of 82 datasets. The five different engines have similar architectures and therefore similar temperature spatial harmonics are expected. Our problem is to estimate the spatial field in engine temperature given a few measurements obtained from thermocouples positioned on a set of rakes. Our motivation for doing so is to understand key engine temperature modes that cannot be captured in a rig or in computational simulations, as the cause of these modes may not be replicated in these simpler environments. To this end, we develop a multivariate linear least-squares model with Tikhonov regularization to estimate the 2D temperature spatial field. Our model uses a Fourier expansion in the circumferential direction and a quadratic polynomial expansion in the radial direction. One important component of our modeling framework is the selection of model parameters, i.e., the harmonics in the circumferential direction. A training-testing paradigm is proposed and applied to quantify the harmonics.

References

1.
Ernst
,
M.
,
Michel
,
A.
, and
Jeschke
,
P.
,
2011
, “
Analysis of Rotor–Stator-Interaction and Blade-to-Blade Measurements in a Two Stage Axial Flow Compressor
,”
ASME J. Turbomach.
,
133
(
1
), p.
011027
. 10.1115/1.4001168
2.
Sanders
,
A.
,
Papalia
,
J.
, and
Fleeter
,
S.
,
2002
, “
Multi-Blade Row Interactions in a Transonic Axial Compressor: Part I—Stator Particle Image Velocimetry (PIV) Investigation
,”
ASME J. Turbomach.
,
124
(
1
), pp.
10
18
. 10.1115/1.1411973
3.
Mailach
,
R.
,
Lehmann
,
I.
, and
Vogeler
,
K.
,
2008
, “
Periodical Unsteady Flow Within a Rotor Blade Row of an Axial Compressor Part I: Flow Field at Midspan
,”
ASME J. Turbomach.
,
130
(
4
), p.
041004
. 10.1115/1.2812329
4.
Stoll
,
F.
,
Tremback
,
J. W.
, and
Arnaiz
,
H. H.
,
1979
, “
Effect of Number of Probes and Their Orientation on the Calculation of Several Compressor Face Distortion Descriptors
,”
Tech. rep., NASA Technical Memorandum 72859
.
5.
Francis
,
S. T.
, and
Morse
,
I. E.
,
1989
,
Measurement and Instrumentation in Engineering: Principles and Basic Laboratory Experiments
, Vol.
67
,
CRC Press
,
Boca Raton, FL
.
6.
Cumpsty
,
N. A.
, and
Horlock
,
J. H.
,
2006
, “
Averaging Nonuniform Flow for a Purpose
,”
ASME J. Turbomach.
,
128
(
1
), pp.
120
129
. 10.1115/1.2098807
7.
Pianko
,
M.
, and
Wazelt
,
F.
,
1983
, “
Suitable Averaging Techniques in Non-Uniform Internal Flows
,”
Tech. rep., AGARD-AR-182
.
8.
Dzung
,
L. S.
,
1971
, “
Consistent Mean Values for Compressible Media in the Theory of Turbomachines
,”
Brown Boveri Rev.
,
58
(
10
), pp.
485
492
.
9.
SAE International
,
2017
, “
Inlet Total-Pressure-Distortion Considerations for Gas-Turbine Engines
,”
Aerospace Information Report, AIR1419(C)
.
10.
Seshadri
,
P.
,
Duncan
,
A.
,
Simpson
,
D.
,
Thorne
,
G.
, and
Parks
,
G.
,
2019
, “
Spatial Flow-Field Approximation Using Few Thermodynamic Measurements Part II: Uncertainty Assessments
,”
ASME J. Turbomach.
, pp.
1
18
. http://dx.doi.org/10.1115/1.4045782
11.
Skiles
,
T. W.
,
1980
, “
Turbine Engine Flowpath Averaging Techniques
,”
Tech. rep
.,
Arnold Air Force Station
.
12.
Chilla
,
M.
,
Pullan
,
G.
, and
Gallimore
,
S.
,
2019
, “
Reducing Instrumentation Errors Caused by Circumferential Flow Field Variations in Multi-Stage Axial Compressors
,”
Proceedings of the ASME Turbo Expo 2019: Turbomachinery Technical Conference and Exposition
,
Phoenix, AZ
,
June 17–21
, p.
V02AT39A018
.
13.
Hastie
,
T.
,
Tibshirani
,
R.
, and
Friedman
,
J.
,
2009
,
The Elements of Statistical Learning
, 2nd ed. (
Series in Statistics
),
Springer
,
New York
.
14.
Rogers
,
S.
, and
Girolami
,
M.
,
2016
,
A First Course in Machine Learning
,
CRC Press
,
Boca Raton, FL
.
15.
Goodfellow
,
I.
,
Bengio
,
Y.
, and
Courville
,
A.
,
2016
,
Deep Learning
,
MIT Press
,
Cambridge, MA
.
16.
Kutner
,
M. H.
,
Nachtsheim
,
C. J.
,
Neter
,
J.
, and
Li
,
W.
,
2005
,
Applied Linear Statistical Models
, Vol.
5
,
McGraw-Hill/Irwin
,
Boston, MA
.
17.
Hansen
,
P. C.
,
2010
,
Discrete Inverse Problems: Insight and Algorithms
(
Fundamentals of Algorithms
),
SIAM
,
Philadelphia, PA
.
18.
Golub
,
G. H.
, and
Van Loan
,
C. F.
,
2013
,
Matrix Computations
, Vol.
4
,
JHU Press
,
Baltimore, MD
.
19.
Castellanos
,
J. L.
,
Gómez
,
S.
, and
Guerra
,
V.
,
2002
, “
The Triangle Method for Finding the Corner of the L-Curve
,”
Appl. Numer. Math.
,
43
(
4
), pp.
359
373
. 10.1016/S0168-9274(01)00179-9
20.
Strang
,
G.
, and
Nguyen
,
T.
,
1996
,
Wavelets and Filter Banks
,
SIAM
,
Philadelphia, PA
.
You do not currently have access to this content.