- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Quaternion structure of azimuthal instabilities

Giulio Ghirardo* and Mirko R. Bothien

  • Ansaldo Energia Switzerland, Baden 5401, Switzerland

  • *Corresponding author: giulio.ghirardo@ansaldoenergia.com
PDF

Phys. Rev. Fluids 3, 113202 – Published 20 November, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.113202

Abstract

Rotationally symmetric systems can exhibit acoustic fluctuations in the azimuthal direction. In experimental works the nature (standing or spinning) of these fluctuations is often described by a set of indicators. These indicators either depend on the chosen frame of reference or are not state space variables for the acoustic field. Conversely, in theoretical works the field is projected on two orthogonal modes, and the system is characterized in terms of two amplitudes and one phase difference. Also in these works the nature of the field is not a state space variable but a derived quantity. Moreover the phase difference between the two modes is undetermined when one of the amplitudes of the two modes is zero, making the phase space ill-posed. We present a solution to these limitations, and we show how the acoustic field can be embedded in quaternion algebra, by calculating a suitable analytic signal of the complex-valued embedding of the acoustic field. This allows us to map the state of the system to a point that moves as function of time on a two-dimensional sphere in three-dimensional (3D) space, the Poincaré-Bloch sphere. To each state of the system corresponds just one point in this 3D space, which is then a well-posed phase space. We term the spherical coordinates of the point the amplitude of oscillation, the nature angle, and the orientation angle. The amplitude of oscillation of the system is the radius of the sphere. The nature angle is the latitude angle and positions the point closer to the equator (pure standing mode states) or closer to the poles (pure spinning mode states). The orientation angle is the longitude angle and describes the position of the pressure antinodes of the part of the acoustic field that is standing. These coordinates have a straightforward physical interpretation, can be easily calculated from experimental data, and are at the same time state space variables of a simple and elegant ansatz that can be used in low-order models. We also present an example of characterization of an experimental azimuthal thermoacoustic instability.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. W. Krebs, G. Walz, P. Flohr, and S. Hoffmann, Modal Analysis of Annular Combustors: Effect of Burner Impedance, in Proc. ASME Turbo Expo 2001 (ASME, New Orleans, LA, 2001), pp. 1–8.
  2. J. Lepers, W. Krebs, B. Prade, P. Flohr, G. Pollarolo, and A. Ferrante, Investigation of thermoacoustic stability limits of an annular gas turbine combustor test-rig with and without Helmholtz-resonators, in Proc. ASME Turbo Expo 2005 (ASME, Reno-Tahoe, NV, 2005), pp. 1–13.
  3. M. R. Bothien, N. Noiray, and B. Schuermans, Analysis of azimuthal thermoacoustic modes in annular gas turbine combustion chambers, in Proc. ASME Turbo Expo 2014 (ASME, Düsseldorf, 2014), pp. 1–11.
  4. J. O'Connor, S. Natarajan, M. Malanoski, and T. Lieuwen, Disturbance Field Characteristics of a Transversely Excited Annular Jet, in Proc. ASME Turbo Expo 2010 (ASME, Glasgow, 2010), pp. 23–38.
  5. J. Schwing, F. Grimm, and T. Sattelmayer, A Model for the Thermo-Acoustic Feedback of Transverse Acoustic Modes and Periodic Oscillations in Flame Position in Cylindrical Flame Tubes, in Proc. ASME Turbo Expo 2012 (ASME, Copenhagen, 2012), p. 14.
  6. G. Ghirardo, C. Di Giovine, J. P. Moeck, and M. R. Bothien, Thermoacoustics of can-annular combustors, J. Eng. Gas Turbines Power 141, 011007 (2018).
  7. J. P. Moeck, M. Paul, and C. O. Paschereit, Thermoacoustic instabilities in an annular Rijke tube, in Proc. ASME Turbo Expo 2010 (ASME, Glasgow, 2010), pp. 1219–1232.
  8. N. Noiray, M. R. Bothien, and B. Schuermans, Investigation of azimuthal staging concepts in annular gas turbines, Combust. Theory Modell. 15, 585 (2011).
  9. M. Bauerheim, M. Cazalens, and T. Poinsot, A theoretical study of mean azimuthal flow and asymmetry effects on thermo-acoustic modes in annular combustors, Proc. Combust. Inst. 35, 3219 (2015).
  10. G. Ghirardo, F. Boudy, and M. R. Bothien, Amplitude statistics prediction in thermoacoustics, J. Fluid Mech. 844, 216 (2018).
  11. L. Crocco, Research on combustion instability in liquid propellant rockets, Symp. (Intl.) Comb. 12, 85 (1969).
  12. G. Ghirardo and M. P. Juniper, Azimuthal instabilities in annular combustors: Standing and spinning modes, Proc. R. Soc. London A 469, 20130232 (2013).
  13. G. Ghirardo, M. P. Juniper, and J. P. Moeck, Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors, J. Fluid Mech. 805, 52 (2016).
  14. N. Noiray and B. Schuermans, On the dynamic nature of azimuthal thermoacoustic modes in annular gas turbine combustion chambers, Proc. R. Soc. London A 469, 20120535 (2013).
  15. J. Flamant, N. Le Bihan, and P. Chainais, Time-frequency analysis of bivariate signals, Appl. Comput. Harmonic Anal. 1, 1 (2017).
  16. J. Flamant, N. Le Bihan, and P. Chainais, Spectral Analysis of Stationary Random Bivariate Signals, IEEE Trans. Signal Proc. 65, 6135 (2017).
  17. T. Bulow and G. Sommer, Hypercomplex signals—A novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Proc. 49, 2844 (2001).
  18. B. Schuermans, C. O. Paschereit, and P. Monkewitz, Non-Linear Combustion Instabilities in Annular Gas-Turbine Combustors, in 44th AIAA Aerospace Sciences Meeting and Exhibit (American Institute of Aeronautics and Astronautics, Reno, NV, 2006), pp. 1–12.
  19. P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Press, Princeton, 1968).
  20. M. L. Munjal, Acoustics of Ducts and Mufflers (John Wiley & Sons, Bangalore, India, 1987).
  21. T. Poinsot, P. Wolf, G. Staffelbach, L. Y. M. Gicquel, and J. D. Muller, Annual Research Briefs, Center for Turbulence Research, Technical report, Center for Turbulence Research, Stanford University (2011).
  22. N. A. Worth and J. R. Dawson, Modal dynamics of self-excited azimuthal instabilities in an annular combustion chamber, Combust. Flame 160, 2476 (2013).
  23. J.-F. Bourgouin, D. Durox, J. P. Moeck, T. Schuller, and S. Candel, Self-sustained instabilities in an annular combustor coupled by azimuthal and longitudinal acoustic modes, in Proc. ASME Turbo Expo 2013 (ASME, San Antonio, TX, 2013), pp. 1–13.
  24. S. Evesque, W. Polifke, and C. Pankiewitz, Spinning and Azimuthally Standing Acoustic Modes in Annular Combustors, in 9th AIAA/CEAS Aeroacoustics Conf. and Exhibit (AIAA, Hilton Head, SC, 2003), pp. 1–8.
  25. N. A. Worth and J. R. Dawson, Effect of equivalence ratio on the modal dynamics of azimuthal combustion instabilities, Proc. Combust. Inst. 36, 3743 (2017).
  26. K. Prieur, D. Durox, T. Schuller, and S. Candel, Strong Azimuthal Combustion Instabilities in a Spray Annular Chamber with Intermittent Partial Blow-Off, J. Eng. Gas Turbines Power 140, 031503 (2017).
  27. J.-F. Bourgouin, D. Durox, J. P. Moeck, T. Schuller, and S. Candel, Characterization and Modeling of a Spinning Thermoacoustic Instability in an Annular Combustor Equipped with Multiple Matrix Injectors, J. Eng. Gas Turbines Power 137, 021503 (2015).
  28. D. Laera, T. Schuller, K. Prieur, D. Durox, S. M. Camporeale, and S. Candel, Flame Describing Function analysis of spinning and standing modes in an annular combustor and comparison with experiments, Combust. Flame 184, 136 (2017).
  29. D. Gabor, Theory of communication, J. Inst. Electr. Eng. 93, 429 (1946).
  30. J. Ville, Théorie et Applications de la Notion de Signal Analytique (translated in English by Michael D. Godfrey in February 2017), Cables Transmissions 2, 61 (1948).
  31. B. Picinbono, On instantaneous amplitude and phase of signals, IEEE Trans. Signal Proc. 45, 552 (1997).
  32. D. Rouwenhorst, J. Hermann, and W. Polifke, Online Monitoring of Thermoacoustic Eigenmodes in Annular Combustion Systems Based on a State-Space Model, J. Eng. Gas Turbines Power 139, 1 (2017).
  33. T. Hummel, F. Berger, B. Schuermans, and T. Sattelmayer, Theory and Modeling of Non-Degenerate Transversal Thermoacoustic Limit Cycle Oscillations, in Intl. Symp.: Thermoacoustic Instabilities in Gas Turbines and Rocket Engines (Munich, 2016), pp. 1–13.
  34. W. R. Hamilton, On Quaternions; or on a new System of Imaginaries in Algebra, Letter to John T. Graves (1843), https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/OnQuat/OnQuat.pdf.
  35. J. Gonella, A rotary-component method for analysing meteorological and oceanographic vector time series, Deep Sea Res. Ocean. Abst. 19, 833 (1972).
  36. P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data (Cambridge University Press, Cambridge, 2010).
  37. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, Cambridge, 2003).
  38. N. Le Bihan, S. J. Sangwine, and T. A. Ell, Instantaneous frequency and amplitude of orthocomplex modulated signals based on quaternion Fourier transform, Signal Process. 94, 308 (2013).
  39. By keeping the slowly varying quantities a,Δ,δ,θ0,φ constant in the limit cycle; see, e.g., discussion around (19) in Ref. [44].
  40. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization (Cambridge University Press, Cambridge, 2001).
  41. A. Balanov, N. Janson, D. Postnov, and O. Sosnovtseva, Synchronization: From Simple to Complex (Springer, Berlin, 2009).
  42. One needs to compare the expressions {Acosχ,Asinχ,nθ0,ωt+φ} of (36) with the expressions {a,b,θ,ϕ} in (18) of Ref. [44].
  43. J. M. Lilly and J.-C. Gascard, Wavelet ridge diagnosis of time-varying elliptical signals with application to an oceanic eddy, Nonlinear Proc. Geophys. 13, 467 (2006).
  44. J. M. Lilly and S. C. Olhede, Bivariate Instantaneous Frequency and Bandwidth, IEEE Trans. Signal Proc. 58, 591 (2010).
  45. C. N. K. Mooers, A technique for the cross spectrum analysis of pairs of complex-valued time series, with emphasis on properties of polarized components and rotational invariants, Deep Sea Res. Ocean. Abst. 20, 1129 (1973).
  46. R. C. Jones, A New Calculus for the Treatment of Optical Systems. Description and Discussion of the Calculus, J. Opt. Soc. Am. 31, 488 (1941).
  47. H. Poincaré, Théorie mathématique de la lumière, Technical report, Faculté des sciences de Paris, Cours de Physique Mathématique (1887).
  48. To be fully accurate, Poincaré chose the angle nθ0 in Eq.  (27) in the domain [−π/2,π/2] instead of [−π,π], and used twice the angle nθ0 as longitude coordinate. This difference is just a technical detail and is discussed towards the end of Appendix pp3.
  49. Indeed, when the system is purely spinning either clockwise or anticlockwise the angle nθ0 is undetermined, in the same manner that at the north and south poles of the sphere the latitude is undetermined.
  50. D. D. Holm, Geometric Mechanics—Part I: Dynamics and Symmetry, 2nd ed. (Imperial College Press, London, 2011).
  51. R. P. Feynman, F. L. Vernon, and R. W. Hellwarth, Geometrical Representation of the Schrödinger Equation for Solving Maser Problems, J. Appl. Phys. 28, 49 (1957).
  52. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. III: Quantum Mechanics (Basic Books, New York, 2011).
  53. J. P. Lowe and K. A. Peterson, Quantum Chemistry, 3rd ed. (Elsevier, Amsterdam, 2006).
  54. P. J. Schreier, Polarization ellipse analysis of nonstationary random signals, IEEE Trans. Signal Proc. 56, 4330 (2008).
  55. P. Rubin-Delanchy and A. T. Walden, Kinematics of complex-valued time series, IEEE Trans. Signal Proc. 56, 4189 (2008).
  56. S. Chandna and A. T. Walden, Simulation methodology for inference on physical parameters of complex vector-valued signals, IEEE Trans. Signal Proc. 61, 5260 (2013).
  57. Technically, these are marginal bivariate PDFs, obtained from the trivariate PDF of the state of the system by marginalizing over one of the three variables.
  58. Projected either on two orthogonal standing modes as in Sec. 3a or on two counter-rotating modes as in Sec. 3b.
  59. G. Rigas, A. R. Oxlade, A. S. Morgans, and J. F. Morrison, Low-dimensional dynamics of a turbulent axisymmetric wake, J. Fluid Mech. 755, R51 (2014).
  60. G. Rigas, L. Esclapez, and L. Magri, Proceedings of the Summer Program 2017 of the Center for Turbulence Research, Technical report, Center for Turbulence Research. Stanford University (2016), arXiv:1703.07405.

Outline

Information

Phys. Rev. Fluids 3, 113202– Published 20 November, 2018
  • Received 17 February 2018
Crossmark - Check for updates button

©2018 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation