Abstract

One of the fundamental tasks in performing robust thermoacoustic design of gas turbine combustors is calculating the modal instability risk, i.e., the probability that a thermoacoustic mode is unstable, given various sources of uncertainty (e.g., operation or boundary conditions). To alleviate the high computational cost associated with conventional Monte Carlo simulation, surrogate modeling techniques are usually employed. Unfortunately, in practice, it is not uncommon that only a small number of training samples can be afforded for surrogate model training. As a result, epistemic uncertainty may be introduced by such an “inaccurate” model, provoking a variation of modal instability risk calculation. In the current study, using Gaussian process (GP) as the surrogate model, we address the following two questions: First, how to quantify the variation of modal instability risk induced by the epistemic surrogate model uncertainty? Second, how to reduce the variation of risk calculation given a limited computational budget for the surrogate model training? For the first question, we leverage on the Bayesian characteristic of the GP model and perform correlated sampling of the GP predictions at different inputs to quantify the uncertainty of risk calculation. We show how this uncertainty shrinks when more training samples are available. For the second question, we adopt an active learning strategy to intelligently allocate training samples such that the trained GP model is highly accurate particularly in the vicinity of the zero growth rate contour. As a result, a more accurate and robust modal instability risk calculation is obtained without increasing the computational cost of surrogate model training.

References

1.
Juniper
,
M. P.
, and
Sujith
,
R. I.
,
2018
, “
Sensitivity and Nonlinearity of Thermoacoustic Oscillations
,”
Annu. Rev. Fluid Mech.
,
50
(
1
), pp.
661
689
.10.1146/annurev-fluid-122316-045125
2.
Guo
,
S.
,
Silva
,
C. F.
,
Ghani
,
A.
, and
Polifke
,
W.
,
2019
, “
Quantification and Propagation of Uncertainties in Identification of Flame Impulse Response for Thermoacoustic Stability Analysis
,”
ASME J. Eng. Gas Turbines Power
,
141
(
2
), p.
021032
.10.1115/1.4041652
3.
Silva
,
C.
,
Magri
,
L.
,
Runte
,
T.
, and
Polifke
,
W.
,
2017
, “
Uncertainty Quantification of Growth Rates of Thermoacoustic Instability by an Adjoint Helmholtz Solver
,”
ASME J. Eng. Gas Turbines Power
,
139
(
1
), p.
011901
.10.1115/1.4034203
4.
Ndiaye
,
A.
,
Bauerheim
,
M.
, and
Nicoud
,
F.
,
2015
, “
Uncertainty Quantification of Thermoacoustic Instabilities on a Swirled Stabilized Combustor
,”
ASME
Paper No. GT2015-44133.10.1115/GT2015-44133
5.
Magri
,
L.
,
Bauerheim
,
M.
,
Nicoud
,
F.
, and
Juniper
,
M. P.
,
2016
, “
Stability Analysis of Thermo-Acoustic Nonlinear Eigenproblems in Annular Combustors. Part II—Uncertainty Quantification
,”
Comput. Phys.
,
325
, pp.
411
421
.10.1016/j.jcp.2016.08.043
6.
Mensah
,
G. A.
,
Magri
,
L.
, and
Moeck
,
J. P.
,
2017
, “
Methods for the Calculation of Thermoacoustic Stability Margins and Monte Carlo-Free Uncertainty Quantification
,”
ASME
Paper No. GT2017-64829.10.1115/GT2017-64829
7.
Avdonin
,
A.
,
Jaensch
,
S.
,
Silva
,
C. F.
,
Češnovar
,
M.
, and
Polifke
,
W.
,
2018
, “
Uncertainty Quantification and Sensitivity Analysis of Thermoacoustic Stability With Non-Intrusive Polynomial Chaos Expansion
,”
Combust. Flame
,
189
, pp.
300
310
.10.1016/j.combustflame.2017.11.001
8.
Avdonin
,
A.
, and
Polifke
,
W.
,
2019
, “
Quantification of the Impact of Uncertainties in Operating Conditions on the Flame Transfer Function With Non-Intrusive Polynomial Chaos Expansion
,”
ASME J. Eng. Gas Turbines Power
,
141
(
1
), p.
011020
.10.1115/1.4040745
9.
Silva
,
C. F.
,
Pettersson
,
P.
,
Iaccarino
,
G.
, and
Ihme
,
M.
,
2018
, “
Uncertainty Quantification of Combustion Noise by Generalized Polynomial Chaos and State-Space Models
,”
Combust. Flame
, pp. 113–130.
10.1016/j.combustflame.2020.03.010
10.
Guo
,
S.
,
Silva
,
C. F.
, and
Polifke
,
W.
,
2019
, “
Efficient Robust Design for Thermoacoustic Instability Analysis: A Gaussian Process Approach
,”
ASME J. Eng. Gas Turbines Power
,
142
(
3
), p.
031026
.10.1115/1.4044197
11.
Balesdent
,
M.
,
Morio
,
J.
, and
Brevault
,
L.
,
2016
, “
Rare Event Probability Estimation in the Presence of Epistemic Uncertainty on Input Probability Distribution Parameters
,”
Method. Comput. Appl. Probab.
,
18
(
1
), pp.
197
216
.10.1007/s11009-014-9411-x
12.
Echard
,
B.
,
Gayton
,
N.
, and
Lemaire
,
M.
,
2011
, “
AK-MCS: An Active Learning Reliability Method Combining Kriging and Monte Carlo Simulation
,”
Struct. Saf.
,
33
(
2
), pp.
145
154
.10.1016/j.strusafe.2011.01.002
13.
Nannapaneni
,
S.
,
Hu
,
Z.
, and
Mahadevan
,
S.
,
2016
, “
Uncertainty Quantification in Reliability Estimation With Limit State Surrogates
,”
Struct. Multidiscip. Optim.
,
54
(
6
), pp.
1509
1526
.10.1007/s00158-016-1487-1
14.
Komarek
,
T.
, and
Polifke
,
W.
,
2010
, “
Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner
,”
ASME J. Eng. Gas Turbines Power
,
132
(
6
), p.
061503
.10.1115/1.4000127
15.
Tay-Wo-Chong
,
L.
,
Bomberg
,
S.
,
Ulhaq
,
A.
,
Komarek
,
T.
, and
Polifke
,
W.
,
2012
, “
Comparative Validation Study on Identification of Premixed Flame Transfer Function
,”
ASME J. Eng. Gas Turbines Power
,
134
(
2
), p.
021502
.10.1115/1.4004183
16.
Oberleithner
,
K.
, and
Paschereit
,
C. O.
,
2016
, “
Modeling Flame Describing Functions Based on Hydrodynamic Linear Stability Analysis
,”
ASME
Paper No. GT2016-57316.10.1115/GT2016-57316
17.
Albayrak
,
A.
,
Juniper
,
M. P.
, and
Polifke
,
W.
,
2019
, “
Propagation Speed of Inertial Waves in Cylindrical Swirling Flows
,”
J. Fluid Mech.
,
879
, pp.
85
120
.10.1017/jfm.2019.641
18.
Jones
,
D. R.
,
Schonlau
,
M.
, and
Welch
,
W. J.
,
1998
, “
Efficient Global Optimization of Expensive Black-Box Functions
,”
J. Global Optim.
,
13
(
4
), pp.
455
492
.10.1023/A:1008306431147
19.
Schneider
,
E.
,
Staudacher
,
S.
,
Schuermans
,
B.
,
Ye
,
H.
, and
Meeuwissen
,
T.
,
2007
, “
Real-Time Modelling of the Thermoacoustic Dynamics of a Gas Turbine Using a Gaussian Process
,”
ASME
Paper No. GT2007-27468. 10.1115/GT2007-27468
20.
Chattopadhyay
,
P.
,
Mondal
,
S.
,
Bhattacharya
,
C.
,
Mukhopadhyay
,
A.
, and
Ray
,
A.
,
2017
, “
Dynamic Data-Driven Design of Lean Premixed Combustors for Thermoacoustically Stable Operations
,”
Mech. Des.
,
139
(
11
), p.
111419
.10.1115/1.4037307
21.
Chattopadhyay
,
P.
,
Mondal
,
S.
,
Ray
,
A.
, and
Mukhopadhyay
,
A.
,
2019
, “
Dynamic Data-Driven Combustor Design for Mitigation of Thermoacoustic Instabilities
,”
ASME J. Dyn. Syst., Meas., Control
,
141
(
1
), p.
014501
.10.1115/1.4040210
22.
Albayrak
,
A.
,
Steinbacher
,
T.
,
Komarek
,
T.
, and
Polifke
,
W.
,
2018
, “
Convective Scaling of Intrinsic Thermo-Acoustic Eigenfrequencies of a Premixed Swirl Combustor
,”
ASME J. Eng. Gas Turbines Power
,
140
(
4
), p.
041510
.10.1115/1.4038083
23.
Echard
,
B.
,
Gayton
,
N.
,
Lemaire
,
M.
, and
Relun
,
N.
,
2013
, “
A Combined Importance Sampling and Kriging Reliability Method for Small Failure Probabilities With Time-Demanding Numerical Models
,”
Reliab. Eng. Syst. Saf.
,
111
, pp.
232
240
.10.1016/j.ress.2012.10.008
24.
Miao
,
F.
, and
Ghosn
,
M.
,
2011
, “
Modified Subset Simulation Method for Reliability Analysis of Structural Systems
,”
Struct. Saf.
,
33
(
4–5
), pp.
251
260
.10.1016/j.strusafe.2011.02.004
25.
Smith
,
R.
,
2014
,
Uncertainty Quantification: Theory, Implementation, and Applications
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
26.
Swiler
,
L.
,
Slepoy
,
R.
, and
Giunta
,
A.
,
2006
, “
Evaluation of Sampling Methods in Constructing Response Surface Approximations
,”
AIAA
Paper No. 2006-1827.10.2514/6.2006-1827
27.
Loeppky
,
J. L.
,
Sacks
,
J.
, and
Welch
,
W. J.
,
2009
, “
Choosing the Sample Size of a Computer Experiment: A Practical Guide
,”
Technometrics
,
51
(
4
), pp.
366
376
.10.1198/TECH.2009.08040
28.
Lieuwen
,
T.
, and
Yang
,
V.
, eds.,
2005
,
Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling
, Vol.
210
, AIAA, Reston, VA.
29.
Palies
,
P.
,
Durox
,
D.
,
Schuller
,
T.
, and
Candel
,
S.
,
2011
, “
Nonlinear Combustion Instability Analysis Based on the Flame Describing Function Applied to Turbulent Premixed Swirling Flames
,”
Combust. Flame
,
158
(
10
), pp.
1980
1991
.10.1016/j.combustflame.2011.02.012
30.
Silva
,
C. F.
,
Nicoud
,
F.
,
Schuller
,
T.
,
Durox
,
D.
, and
Candel
,
S.
,
2013
, “
Combining a Helmholtz Solver With the Flame Describing Function to Assess Combustion Instability in a Premixed Swirled Combustor
,”
Combust. Flame
,
160
(
9
), pp.
1743
1754
.10.1016/j.combustflame.2013.03.020
You do not currently have access to this content.