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Research Papers: Gas Turbines: Combustion, Fuels, and Emissions

# Reduced-Order Modeling of Aeroacoustic Systems for Stability Analyses of Thermoacoustically Noncompact Gas Turbine Combustors

[+] Author and Article Information
Tobias Hummel

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching D-85748, Germany;
Technische Universität München,
Garching D-85748, Germany
e-mail: hummel@td.mw.tum.de

Constanze Temmler

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching D-85748, Germany
e-mail: temmler@td.mw.tum.de

Bruno Schuermans

Technische Universität München,
Garching D-85748, Germany;
Alstom Power,
e-mail: bruno.schuermans@power.alstom.com

Thomas Sattelmayer

Lehrstuhl für Thermodynamik,
Technische Universität München,
Garching D-85748, Germany
e-mail: sattelmayer@td.mw.tum.de

1Corresponding author.

Contributed by the Coal, Biomass and Alternate Fuels Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 17, 2015; final manuscript received August 28, 2015; published online October 27, 2015. Editor: David Wisler.

J. Eng. Gas Turbines Power 138(5), 051502 (Oct 27, 2015) (11 pages) Paper No: GTP-15-1342; doi: 10.1115/1.4031542 History: Received July 17, 2015; Revised August 28, 2015

## Abstract

A methodology is presented to model noncompact thermoacoustic phenomena using reduced-order models (ROMs) based on the linearized Navier–Stokes equations (LNSEs). The method is applicable to geometries with a complex flow field as in a gas turbine combustion chamber. The LNSEs, and thus the resulting ROM, include coupling effects between acoustics and mean fluid flow and are hence capable of describing propagation and (e.g., vortical) damping of the acoustic fluctuations within the considered volume. Such an ROM then constitutes the main building block for a novel thermoacoustic stability analysis method via a low-order hybrid approach. This method presents an expansion to state-of-the-art low-order stability tools and is conceptually based on three core features: First, the multidimensional and volumetric nature of the ROM establishes access to account spatial variability and noncompact effects on heat-release fluctuations. As a result, it is particularly useful for high-frequency phenomena such as screech. Second, the LNSE basis grants the ROM the capability to reconstruct complex acoustic performances physically accurate. Third, the formulation of the ROM in state-space allows convenient access to the frequency and time domain. In the time domain, nonlinear saturation mechanisms can be included, which reproduce the nonlinear stochastic limit cycle behavior of thermoacoustic oscillations. In order to demonstrate and verify the ROM's underlying methodology, a test case using an orifice-tube geometry as the acoustic volume is performed. The generation of the ROM of the orifice tube is conducted in a two-step procedure. As the first step, the geometrical domain is aeroacoustically characterized through the LNSE in frequency domain and discretized via the finite element method (FEM). The second step concerns the actual derivation of the ROM. The high-order dynamical system from the LNSE discretization is subjected to a modal reduction as order reduction technique. Mathematically, this modal reduction is the projection of the high-order ($N∼$ 200,000) system into its truncated left eigenspace. An order reduction of several magnitudes (ROM order: $Nr∼$ 100) is achieved. The resulting ROM contains all essential information about propagation and damping of the acoustic variables, and efficiently reproduces the aeroacoustic performance of the orifice tube. Validation is achieved by comparing ROM results against numerical and experimental benchmarks from LNSE–FEM simulations and test rig measurements, respectively. Excellent agreement is found, which grants the ROM modeling approach full eligibility for further usage in the context of thermoacoustic stability modeling. This work is concluded by a methodological demonstration of performing stability analyses of noncompact thermoacoustic systems using the herein presented ROMs.

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## References

Sattelmayer, T. , 2010, “ Stationäre Gasturbinen, 2. neu bearbeitete Auflage,” Grundlagen der Verbrennung in Stationären Gasturbinen, Springer-Verlag, Heidelberg, Germany, pp. 397–452.
Sattelmayer, T. , 2000, “ Influence of the Combustor Aerodynamics on Combustion Instabilities From Equivalence Ratio Fluctuations,” ASME Paper No. GT2000-0082.
Schuermans, B. , Bellucci, V. , and Paschereit, C. O. , 2003, “ Thermoacoustic Modeling and Control of Multi Burner Combustion Systems,” ASME Paper No. GT2005-38688.
Emmert, T. , Jaensch, S. , Sovardi, C. , and Polifke, W. , 2014, “ taX—A Flexible Tool for Low-Order Duct Acoustic Simulation in Time and Frequency Domain,” Forum Acusticum, Krakow, Poland, Sept. 7–12.
Schuermans, B. , Bellucci, V. , Guethe, F. , Meili, F. , Flohr, P. , and Paschereit, C. O. , 2004, “ A Detailed Analysis of Thermoacoustic Interaction Mechanisms in a Turbulent Premixed Flame,” ASME Paper No. GT2004-53831.
Schuermans, B. , Guethe, F. , and Mohr, W. , 2010, “ Optical Transfer Function Measurements for Technically Premixed Flames,” ASME J. Eng. Gas Turbines Power, 132(8), p. 081501.
Paschereit, C. O. , Schuermans, B. , Polifke, W. , and Mattson, O. , 2002, “ Measurement of Transfer Matrices and Source Terms of Premixed Flames,” ASME J. Eng. Gas Turbines Power, 124(2), pp. 239–247.
Polifke, W. , 2014, “ Black-Box System Identification for Reduced Order Model Construction,” Ann. Nucl. Energy, 67, pp. 109–128.
Polifke, W. , 2004, Combustion Instabilities (VKI Lecture Series), von Karman Institute for Fluid Dynamics, Rhode-St-Genèse, Belgium.
Schuermans, B. , Paschereit, C. O. , and Monkewitz, P. , 2006, “ Non-Linear Combustion Instabilities in Annular Gas-Turbine Combustors,” AIAA Paper No. 2006-549.
Pieringer, J. E. , 2008, “ Simulation Selbsterregter Verbrennungsschwingungen in Raketenschubkammern im Zeitbereich,” Ph.D. thesis, Lehrstuhl f. Thermodynamik, Technische Universität München, Munich, Germany.
Schwing, J. , Grimm, F. , and Sattelmayer, T. , 2012, “ A Model for Thermo-Acoustic Feedback of Transverse Acoustic Modes and Periodic Oscillations in Flame Positions in Cylindrical Flame Tubes,” ASME Paper No. GT2012-68775.
Schwing, J. , and Sattelmayer, T. , 2013, “ High-Frequency Instabilities in Cylindrical Flame Tubes: Feedback Mechanism and Damping,” ASME Paper No. GT2013-94064.
Zellhuber, M. , Schwing, J. , Schuermans, B. , Sattelmayer, T. , and Polifke, W. , 2014, “ Experimental and Numerical Investigations of Thermoacoustic Sources Related to High-Frequency Instabilities,” Int. J. Spray Combust. Dyn., 6(1), pp. 1–34.
Zellhuber, M. , 2013, “ High Frequency Response of Auto-Ignition and Heat Release to Acoustic Perturbations,” Ph.D. thesis, Lehrstuhl f. Thermodynamik, Technische Universität München, Munich, Germany.
Culick, F. E. C. , 2006, “ Unsteady Motions in Combustion Chambers for Propulsion Systems,” NATO Advisory Group for Aerospace Research and Development, Neuilly-sur-Seine, France, RTO AGARDograph AG-AVT-039, No. AC/323(AVT-039)TP/103.
Lieuwen, T. , 2012, Unsteady Combustor Physics, Cambridge University Press, New York.
Gikadi, J. , Föller, S. , and Sattelmayer, T. , 2014, “ Impact of Turbulence on the Prediction of Linear Acoustic Interactions: Acoustic Response of a Turbulent Shear Layer,” J. Sound Vib., 333(24), pp. 6548–6559.
Rao, P. , and Morris, P. , 2006, “ Use of Finite Element Methods in Frequency Domain Aeroacoustics,” AIAA J., 44(7), pp. 1643–1652.
Gikadi, J. , Schulze, M. , Föller, S. , Schwing, J. , and Sattelmayer, T. , 2012, “ Linearized Navier–Stokes and Euler Equations for the Determination of the Acoustic Scattering Behaviour of and Area Expansion,” AIAA Paper No. 2012-2292.
Gikadi, J. , 2013, “ Prediction of Acoustic Modes in Combustors Using Linearized Navier–Stokes Equations in Frequency Space,” Ph.D. thesis, Lehrstuhl f. Thermodynamik, Technische Universität München, Munich, Germany.
Schulze, M. , and Sattelmayer, T. , 2014, “ Time and Frequency Domain Descriptions of Thermoacoustics in Rocket Engines With Focus on Dome Coupling—A Comparison,” Universität München, Munich, Germany, Sonderforschungsbereich/Transregio 40 Annual Report.
Bothien, M. R. , Noiray, N. , and Schuermans, B. , 2014, “ Analysis of Azimuthal Thermoacoustic Modes in Annular Gas Turbine Combustion Chambers,” ASME J. Eng. Gas Turbines Power, 137(6), p. 061505.
Benner, P. , 2011, “ Model Reduction for Linear Dynamical Systems,” Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, Trogir, Croatia, Oct. 10–15.
Antoulas, A. C. , 2005, Approximation of Large-Scale Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, PA.
Schulze, M. , Wagner, M. , and Sattelmayer, T. , 2013, “ Acoustic Scattering Properties of Perforated Plates and Orifices With Stratified Flow-Hf-8 Test Case,” 3rd REST Modelling Workshop, Vernon, CA, Mar. 27–28 .
Abom, M. , 1991, “ Measurement of the Scattering-Matrix of Acoustical Two-Port,” Mech. Syst. Signal Process., 5(2), pp. 89–104.
Abom, M. , 1992, “ A Note on the Experimental Determination of Acoustical Two-Port Matrices,” J. Sound Vib., 155(1), pp. 185–188.
Ullrich, W. C. , Gikadi, J. , Jörg, C. , and Sattelmayer, T. , 2014, “ Acoustic-Entropy Coupling Behavior and Acoustic Scattering Properties of a Laval Nozzle,” AIAA Paper No. 2014-3193.
ANSYS, General Documentation—Fluent, Release 13, ANSYS, Inc., Canonsburg, PA.
Schuermans, B. , Luebcke, H. , Bajusz, D. , and Flohr, P. , 2005, “ Thermoacoustic Analysis of Gas Turbine Combustion Systems Using Unsteady CFD,” ASME Paper No. GT2005-68393.

## Figures

Fig. 1

Orifice-tube configuration

Fig. 2

Mesh refinement at orifice

Fig. 3

Numerical setup of the orifice tube

Fig. 4

Scattering matrix

Fig. 5

Pressure responses: (a) upstream response–upstream excitation, (b) downstream response–upstream excitation, (c) upstream response–downstream excitation, and (d) downstream response–downstream excitation

Fig. 6

Scattering matrix coefficients: (a) amplitudes and (b) phase angles

Fig. 7

Temperature field (K) and excitation information

Fig. 8

ROM verification responses: (a) L-excitation, (b) T1-excitation, and (c) T2-excitation

Fig. 9

Schematic—flame subregions

Fig. 10

Schematic—MIMO feedback

Fig. 11

Growth rates

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