Skip to main content

Advertisement

SpringerLink
Log in

Optimisation of chaotically perturbed acoustic limit cycles

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure. The objective of manufacturers is to design stable thermoacoustic configurations. In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation. In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down. First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles. Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time-averaged cost functionals exist and can be robustly calculated by a shadowing method. Third, we calculate the sensitivities of the time-averaged acoustic energy and Rayleigh index to small changes to the heat-source intensity and time delay. By embedding the sensitivities into a gradient-update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source. The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. For example, in large-eddy simulations, the nonlinear operator \({\varvec{F}}\) in the dynamical system formulation in Eq. (4) is provided by the spatial discretisation of the momentum, mass and energy conservation laws and chemistry equations [63].

  2. It has been hypothesised by Gallavotti and Cohen [68], Gallavotti [69] that most high-dimensional physical systems develop asymptotically on an attracting set, the dynamics of which can be regarded as hyperbolic. This is called the chaotic hypothesis, which stems from measure theory of turbulence of [70].

  3. This paper provides a method to optimise deterministic thermoacoustic systems. Including stochastic processes in the optimisation of chaotic thermoacoustic systems is beyond the scope of this paper and is left for future work.

References

  1. Lieuwen, T.C.,Yang, V.: Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling. American Institute of Aeronautics and Astronautics, Inc. (2005). ISBN 156347669X. http://www.lavoisier.fr/livre/notice.asp?id=RAXW2KALXX6OWV

  2. Culick, F.E.C.: Unsteady motions in combustion chambers for propulsion systems. RTO AGARDograph AG-AVT-039, North Atlantic Treaty Organization (2006). ISBN 9789283700593

  3. Dowling, A.P., Mahmoudi, Y.: Combustion noise. Proc. Combust. Inst. 35(1), 65–100 (2015). https://doi.org/10.1016/j.proci.2014.08.016

    Article  Google Scholar 

  4. Poinsot, T.: Prediction and control of combustion instabilities in real engines. Proc. Combust. Inst. 36(1), 1–28 (2017). https://doi.org/10.1016/j.proci.2016.05.007

    Article  MathSciNet  Google Scholar 

  5. Juniper, M.P., Sujith, R.: Sensitivity and nonlinearity of thermoacoustic oscillations. Annu. Rev. Fluid Mech. 50(1), 661–689 (2018). https://doi.org/10.1146/annurev-fluid-122316-045125

    Article  MathSciNet  MATH  Google Scholar 

  6. Rayleigh, L.: The explanation of certain acoustical phenomena. Nature 18, 319 (1878). https://doi.org/10.1038/018319a0

    Article  Google Scholar 

  7. Kabiraj, L., Sujith, R.I., Wahi, P.: Bifurcations of self-excited ducted Laminar premixed flames. J. Eng. Gas Turbines Power 134(3), 31502 (2011). https://doi.org/10.1115/1.4004402

    Article  Google Scholar 

  8. Gotoda, H., Nikimoto, H., Miyano, T., Tachibana, S.: Dynamic properties of combustion instability in a lean premixed gas-turbine combustor. Chaos 21(1), 013124 (2011). https://doi.org/10.1063/1.3563577

    Article  Google Scholar 

  9. Gotoda, H., Ikawa, T., Maki, K., Miyano, T.: Short-term prediction of dynamical behavior of flame front instability induced by radiative heat loss. Chaos 22(1), 033106 (2012). https://doi.org/10.1063/1.4731267

    Article  Google Scholar 

  10. Kabiraj, L., Saurabh, A., Wahi, P., Sujith, R.I.: Route to chaos for combustion instability in ducted laminar premixed flames. Chaos 22(2), 023129 (2012). https://doi.org/10.1063/1.4718725

    Article  Google Scholar 

  11. Kashinath, K., Waugh, I.C., Juniper, M.P.: Nonlinear self-excited thermoacoustic oscillations of a ducted premixed flame: bifurcations and routes to chaos. J. Fluid Mech. 761, 399–430 (2014). https://doi.org/10.1017/jfm.2014.601

    Article  MathSciNet  Google Scholar 

  12. Waugh, I.C., Kashinath, K., Juniper, M.P.: Matrix-free continuation of limit cycles and their bifurcations for a ducted premixed flame. J. Fluid Mech. 759, 1–27 (2014). https://doi.org/10.1017/jfm.2014.549

    Article  MathSciNet  Google Scholar 

  13. Nair, V., Thampi, G., Sujith, R.I.: Intermittency route to thermoacoustic instability in turbulent combustors. J. Fluid Mech. 756, 470–487 (2014). https://doi.org/10.1017/jfm.2014.468

    Article  Google Scholar 

  14. Orchini, A., Illingworth, S.J., Juniper, M.P.: Frequency domain and time domain analysis of thermoacoustic oscillations with wave-based acoustics. J. Fluid Mech. 775, 387–414 (2015). https://doi.org/10.1017/jfm.2015.139

    Article  MathSciNet  MATH  Google Scholar 

  15. Nair, V., Sujith, R.I.: A reduced-order model for the onset of combustion instability: physical mechanisms for intermittency and precursors. Proc. Combust. Inst. 35(3), 3193–3200 (2015). https://doi.org/10.1016/j.proci.2014.07.007

    Article  Google Scholar 

  16. Dowling, A.P.: Nonlinear self-excited oscillations of a ducted flame. J. Fluid Mech. 346, 271–290 (1997). https://doi.org/10.1017/S0022112097006484

    Article  MathSciNet  MATH  Google Scholar 

  17. Dowling, A.P.: A kinematic model of a ducted flame. J. Fluid Mech. 394, 51–72 (1999). https://doi.org/10.1017/S0022112099005686

    Article  MATH  Google Scholar 

  18. Eckmann, J.-P.: Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53(4), 643–654 (1981)

    Article  MathSciNet  Google Scholar 

  19. Miles, J.: Strange attractors in fluid dynamics. Adv. Appl. Mech. 24, 189–214 (1984)

    Article  MathSciNet  Google Scholar 

  20. Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985). https://doi.org/10.1103/RevModPhys.57.617

    Article  MathSciNet  MATH  Google Scholar 

  21. Huhn, F., Magri, L.: Stability, sensitivity and optimisation of chaotic acoustic oscillations. J. Fluid Mech. 882, A24 (2020). https://doi.org/10.1017/jfm.2019.828

    Article  MathSciNet  MATH  Google Scholar 

  22. Lieuwen, T.: Unsteady Combustor Physics. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  23. Noiray, N., Schuermans, B.: Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors. Int. J. Non Linear Mech. 50, 152–163 (2013). https://doi.org/10.1016/j.ijnonlinmec.2012.11.008

    Article  Google Scholar 

  24. Noiray, N.: Linear growth rate estimation from dynamics and statistics of acoustic signal envelope in turbulent combustors. J. Eng. Gas Turbines Power 139(4), 041503 (2016). https://doi.org/10.1115/1.4034601

    Article  Google Scholar 

  25. Noiray, N., Denisov, A.: A method to identify thermoacoustic growth rates in combustion chambers from dynamic pressure time series. Proc. Combust. Inst. 36(3), 3843–3850 (2017). https://doi.org/10.1016/j.proci.2016.06.092

    Article  Google Scholar 

  26. Matveev, K.I., Culick, F.E.C.: A model for combustion instability involving vortex shedding. Combust. Sci. Technol. 175, 1059–1083 (2003)

    Article  Google Scholar 

  27. Duncan Thompson, P.: Uncertainty of initial state as a factor in the predictability of large scale atmospheric flow patterns. Tellus A 9, 275–295 (1957). https://doi.org/10.3402/tellusa.v9i3.9111

    Article  MathSciNet  Google Scholar 

  28. Deissler, R.G.: Is Navier–Stokes turbulence chaotic? Phys. Fluids 29(5), 1453 (1986). https://doi.org/10.1063/1.865663

    Article  MathSciNet  MATH  Google Scholar 

  29. Boffetta, G., Cencini, M., Falcioni, M., Vulpiani, A.: Predictability: a way to characterize complexity. Phys. Rep. 356, 367–474 (2002). https://doi.org/10.1016/S0370-1573(01)00025-4

    Article  MathSciNet  MATH  Google Scholar 

  30. Nastac, G., Labahn, J., Magri, L., Ihme, M.: Lyapunov exponent as a metric for assessing the dynamic content and predictability of large-eddy simulations. Phys. Rev. Fluids 2(9), 094606 (2017). https://doi.org/10.1103/PhysRevFluids.2.094606

    Article  Google Scholar 

  31. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963). https://doi.org/10.1175/1520-0469(1963)020\(<\)0130:DNF\(>\)2.0.CO;2

  32. Selle, L., Blouquin, R., Théron, M., Dorey, L.H., Schmid, M., Anderson, W.: Prediction and analysis of combustion instabilities in a model rocket engine. J. Propuls. Power 30(4), 978–990 (2014). https://doi.org/10.2514/1.B35146

    Article  Google Scholar 

  33. Ghani, A., Poinsot, T., Gicquel, L., Müller, J.D.: LES study of transverse acoustic instabilities in a swirled kerosene/air combustion chamber. Flow Turbul. Combust. 96(1), 207–226 (2016). https://doi.org/10.1007/s10494-015-9654-9

    Article  Google Scholar 

  34. Brien, J.O., Kim, J., Ihme, M.: Investigation of the mechanisms of jet-engine core noise using large-eddy simulation. AIAA 54th Aerospace Sciences Meeting, pp. 2016-0761 (2016). https://doi.org/10.2514/6.2016-0761

  35. Gotoda, H., Okuno, Y., Hayashi, K., Tachibana, S.: Characterization of degeneration process in combustion instability based on dynamical systems theory. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 92(5), 1–11 (2015). https://doi.org/10.1103/PhysRevE.92.052906

    Article  Google Scholar 

  36. Unni, V.R., Sujith, R.I.: Multifractal characteristics of combustor dynamics close to lean blowout. J. Fluid Mech. 784, 30–50 (2015). https://doi.org/10.1017/jfm.2015.567

    Article  Google Scholar 

  37. Murugesan, M., Sujith, R.I.: Combustion noise is scale-free: transition from scale-free to order at the onset of thermoacoustic instability. J. Fluid Mech. 772, 225–245 (2015). https://doi.org/10.1017/jfm.2015.215

    Article  Google Scholar 

  38. Pawar, S.A., Seshadri, A., Unni, V.R., Sujith, R.I.: Thermoacoustic instability as mutual synchronization between the acoustic field of the confinement and turbulent reactive flow. J. Fluid Mech. 827, 664–693 (2017). https://doi.org/10.1017/jfm.2017.438

    Article  MathSciNet  MATH  Google Scholar 

  39. Mondal, S., Unni, V.R., Sujith, R.I.: Onset of thermoacoustic instability in turbulent combustors: An emergence of synchronized periodicity through formation of chimera-like states. J. Fluid Mech. 811, 659–681 (2017). https://doi.org/10.1017/jfm.2016.770

    Article  MathSciNet  MATH  Google Scholar 

  40. Magri, L.: Adjoint methods as design tools in thermoacoustics. Appl. Mech. Rev. (2019). https://doi.org/10.1115/1.4042821

    Article  Google Scholar 

  41. Magri, L., Juniper, M.P.: Global modes, receptivity, and sensitivity analysis of diffusion flames coupled with duct acoustics. J. Fluid Mech. 752, 237–265 (2014). https://doi.org/10.1017/jfm.2014.328

    Article  MathSciNet  MATH  Google Scholar 

  42. Magri, L., Juniper, M.P.: Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach. J. Fluid Mech. 719, 183–202 (2013). https://doi.org/10.1017/jfm.2012.639

    Article  MathSciNet  MATH  Google Scholar 

  43. Magri, L., Bauerheim, M., Nicoud, F., Juniper, M.P.: Stability analysis of thermo-acoustic nonlinear eigenproblems in annular combustors. Part II. Uncertainty quantification. J. Comput. Phys. 325, 411–421 (2016). https://doi.org/10.1016/j.jcp.2016.08.043

    Article  MathSciNet  MATH  Google Scholar 

  44. Silva, C.F., Magri, L., Runte, T., Polifke, W.: Uncertainty quantification of growth rates of thermoacoustic instability by an adjoint Helmholtz solver. J. Eng. Gas Turbines Power 139(1), 011901 (2016). https://doi.org/10.1115/1.4034203

    Article  Google Scholar 

  45. Mensah, G.A., Magri, L., Moeck, J.P.: Methods for the calculation of thermoacoustic stability margins and Monte-Carlo free uncertainty quantification. J. Eng. Gas Turbines Power 140(6), 061501 (2018). https://doi.org/10.1115/1.4038156

    Article  Google Scholar 

  46. Mensah, G.A., Moeck, J.P.: Acoustic damper placement and tuning for annular combustors: an adjoint-based optimization study. J. Eng. Gas Turbines Power 139(6), 061501 (2017). https://doi.org/10.1115/1.4035201

    Article  Google Scholar 

  47. Aguilar, J., Juniper, M.P.: Adjoint methods for elimination of thermoacoustic oscillations in a model annular combustor via small geometry modifications. In: ASME Turbo Expo, Oslo, Norway, pp. GT2018–75692 (2018)

  48. Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Nat. Acad. Sci. USA 17(12), 656–660 (1931)

    Article  Google Scholar 

  49. Lea, D.J., Allen, M.R., Haine, T.W.: Sensitivity analysis of the climate of a chaotic system. Tellus A Dyn. Meteorol. Oceanogr. 52(5), 523–532 (2000). https://doi.org/10.3402/tellusa.v52i5.12283

    Article  Google Scholar 

  50. Eyink, G.L., Haine, T.W., Lea, D.J.: Ruelle’s linear response formula, ensemble adjoint schemes and Lévy flights. Nonlinearity 17(5), 1867–1889 (2004). https://doi.org/10.1088/0951-7715/17/5/016

    Article  MathSciNet  MATH  Google Scholar 

  51. Wang, Q.: Forward and adjoint sensitivity computation of chaotic dynamical systems. J. Comput. Phys. 235, 1–13 (2013). https://doi.org/10.1016/j.jcp.2012.09.007

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang, Q., Hu, R., Blonigan, P.: Least squares shadowing sensitivity analysis of chaotic limit cycle oscillations. J. Comput. Phys. 267, 210–224 (2014). https://doi.org/10.1016/j.jcp.2014.03.002

    Article  MathSciNet  MATH  Google Scholar 

  53. Ni, A., Wang, Q.: Sensitivity analysis on chaotic dynamical systems by non-intrusive least squares shadowing (nilss). J. Comput. Phys. 347, 56–77 (2017). https://doi.org/10.1016/j.jcp.2017.06.033

    Article  MathSciNet  MATH  Google Scholar 

  54. Blonigan, P.J., Wang, Q.: Multiple shooting shadowing for sensitivity analysis of chaotic dynamical systems. J. Comput. Phys. 354, 447–475 (2018). https://doi.org/10.1016/j.jcp.2017.10.032

    Article  MathSciNet  MATH  Google Scholar 

  55. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  56. Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996). https://doi.org/10.1017/CBO9780511622700

    Book  MATH  Google Scholar 

  57. Pilyugin, S.Y.: Shadowing in Dynamical Systems. Springer, Berlin (2006)

    MATH  Google Scholar 

  58. Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22(4), 855 (2009)

    Article  MathSciNet  Google Scholar 

  59. Ginelli, F., Poggi, P., Turchi, A., Chaté, H., Livi, R., Politi, A.: Characterizing dynamics with covariant lyapunov vectors. Phys. Rev. Lett. 99(13), 130601 (2007). https://doi.org/10.1103/PhysRevLett.99.130601

    Article  Google Scholar 

  60. Ginelli, F., Chaté, H., Livi, R., Politi, A.: Covariant lyapunov vectors. J. Phys. A Math. Theor. 46(25), 254005 (2013)

    Article  MathSciNet  Google Scholar 

  61. Takeuchi, K.A., Yang, H.-L., Ginelli, F., Radons, G., Chaté, H.: Hyperbolic decoupling of tangent space and effective dimension of dissipative systems. Phys. Rev. E 84, 046214 (2011). https://doi.org/10.1103/PhysRevE.84.046214

    Article  Google Scholar 

  62. Oseledets, V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968)

    MATH  Google Scholar 

  63. Veynante, D., Vervisch, L.: Turbulent combustion modeling. Prog. Energy Combust. Sci. 28, 193–266 (2002). https://doi.org/10.1016/S0360-1285(01)00017-X

    Article  Google Scholar 

  64. Fr\(\varnothing \)yland, J., Alfsen, K.H.: Lyapunov-exponent spectra for the Lorenz model. Phys. Rev. A 29(5), 2928–2931 (1984). https://doi.org/10.1103/PhysRevA.29.2928

  65. Araujo, V., Pacifico, M.J.: Three-Dimensional Flows. A Series of Modern Surveys in Mathematics, vol. 53. Springer, New York (2010). ISBN 9783642263804

  66. Araujo, V., Galatolo, S., Pacifico, M.J.: Statistical properties of Lorenz like flows, recent developments and perspectives. Int. J. Bifurc. Chaos 24(10), 1430028-1–1430028-34 (2014). https://doi.org/10.1142/S0218127414300286

    Article  MathSciNet  MATH  Google Scholar 

  67. Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Applied Mathematical Sciences, vol. 41, 1 st edn. Springer, New York (1982). https://doi.org/10.1007/978-1-4612-5767-7. ISBN 978-1-4612-5767-7

  68. Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80(5–6), 931–970 (1995). https://doi.org/10.1007/BF02179860

    Article  MathSciNet  MATH  Google Scholar 

  69. Gallavotti, G.: Entropy, thermostats, and chaotic hypothesis. Chaos Interdiscip. J. Nonlinear Sci. 16(4), 043114 (2006). https://doi.org/10.1063/1.2372713

    Article  MathSciNet  MATH  Google Scholar 

  70. Ruelle, D.: Measures describing a turbulent flow. Ann. NY Acad. Sci. 357(1), 1–9 (1980). https://doi.org/10.1111/j.1749-6632.1980.tb29669.x

    Article  MATH  Google Scholar 

  71. Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74(2), 189–197 (1980). https://doi.org/10.1007/BF01197757

    Article  MathSciNet  Google Scholar 

  72. Aurell, E., Boffetta, G., Crisanti, A., Paladin, G., Vulpiani, A.: Growth of non-infinitesimal perturbations in turbulence. Phys. Rev. Lett. 77(7), 1262–1265 (1996). https://doi.org/10.1103/PhysRevLett.77.1262

    Article  MATH  Google Scholar 

  73. Balasubramanian, K., Sujith, R.I.: Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20(4), 044103 (2008). https://doi.org/10.1063/1.2895634

    Article  MATH  Google Scholar 

  74. Juniper, M.P.: Triggering in the horizontal rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272–308 (2011). https://doi.org/10.1017/S0022112010004453

    Article  MATH  Google Scholar 

  75. King, L.V.: On the convection of heat from small cyclinders in a stream of fluid: determination of the convection constants of small platinum wires with applications to hot-wire anemometry. Proc. R. Soc. Lond. 214(8), 373–434 (1914)

  76. Heckl, M.A.: Active control of the noise from a Rijke tube. J. Sound Vib. 124(1), 117–133 (1988). https://doi.org/10.1016/S0022-460X(88)81408-1

    Article  Google Scholar 

  77. Heckl, M.A.: Non-linear acoustic effects in the Rijke tube. Acustica 72, 63–71 (1990)

    Google Scholar 

  78. Polifke, W., Poncet, A., Paschereit, C.O., Döbbeling, K.: Reconstruction of acoustic transfer matrices by instationary computational fluid dynamics. J. Sound Vib. 245(3), 483–510 (2001). https://doi.org/10.1006/jsvi.2001.3594

    Article  Google Scholar 

  79. Orchini, A., Rigas, G., Juniper, M.P.: Weakly nonlinear analysis of thermoacoustic bifurcations in the rijke tube. J. Fluid Mech. 805, 523–550 (2016). https://doi.org/10.1017/jfm.2016.585

    Article  MathSciNet  MATH  Google Scholar 

  80. Kashinath, K., Hemchandra, S., Juniper, M.P.: Nonlinear phenomena in thermoacoustic systems with premixed flames. J. Eng. Gas Turbines Power 135(6), 061502 (2013). https://doi.org/10.1115/1.4023305

    Article  Google Scholar 

  81. Kashinath, K., Hemchandra, S., Juniper, M.P.: Nonlinear thermoacoustics of ducted premixed flames: the influence of perturbation convection speed. Combust. Flame 160(12), 2856–2865 (2013). https://doi.org/10.1016/j.combustflame.2013.06.019

    Article  Google Scholar 

  82. Tyagi, M., Jamadar, N., Chakravarthy, S.: Oscillatory response of an idealized two-dimensional diffusion flame: analytical and numerical study. Combust. Flame 149(3), 271–285 (2007). https://doi.org/10.1016/j.combustflame.2006.12.020

    Article  Google Scholar 

  83. Magri, L., See, Y.-C., Tammisola, O., Ihme, M., Juniper, M.: Multiple-scale thermo-acoustic stability analysis of a coaxial jet combustor. Proc. Combust. Inst. (2017). https://doi.org/10.1016/j.proci.2016.06.009

    Article  Google Scholar 

  84. Zinn, B.T., Lores, M.E.: Application of the Galerkin method in the solution of non-linear axial combustion instability problems in liquid rockets. Combust. Sci. Technol. 4(1), 269–278 (1971). https://doi.org/10.1080/00102207108952493

    Article  Google Scholar 

  85. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics, 2nd edn. Pergamon Press, Oxford (1987)

    Google Scholar 

  86. Dowling, A.P., Morgans, A.S.: Feedback control of combustion oscillations. Annu. Rev. Fluid Mech. 37, 151–182 (2005). https://doi.org/10.1146/annurev.fluid.36.050802.122038

    Article  MathSciNet  MATH  Google Scholar 

  87. Trefethen, L.N.: Spectral Methods in MATLAB, vol. 10. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  88. Kennedy, C.A., Carpenter, M.H., Lewis, R.: Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations. Appl. Numer. Math. 35(3), 177–219 (2000). https://doi.org/10.1016/S0168-9274(99)00141-5

    Article  MathSciNet  MATH  Google Scholar 

  89. Ni, A.: Hyperbolicity, shadowing directions and sensitivity analysis of a turbulent three-dimensional flow. J. Fluid Mech. 863, 644–669 (2019). https://doi.org/10.1017/jfm.2018.986

    Article  MathSciNet  MATH  Google Scholar 

  90. Blonigan, P.J., Fernandez, P., Murman, S.M., Wang, Q., Rigas, G., Magri, L.: Towards a Chaotic Adjoint for LES. Center for Turbulence Research, Summer Program (2016)

  91. Fernandez, P., Wang, Q.: Lyapunov spectrum of the separated flow around the NACA 0012 airfoil and its dependence on numerical discretization. J. Comput. Phys. 350, 453–469 (2017). https://doi.org/10.1016/j.jcp.2017.08.056

    Article  MathSciNet  MATH  Google Scholar 

  92. Chu, B.T.: On the energy transfer to small disturbances in fluid flow (Part I). Acta Mech. 1(3), 215–234 (1965)

    Article  Google Scholar 

  93. George, K.J., Sujith, R.: Disturbance energy norms: a critical analysis. J. Sound Vib. 331(7), 1552–1566 (2012). https://doi.org/10.1016/j.jsv.2011.11.027

    Article  Google Scholar 

  94. Magri, L.: Adjoint methods in thermo-acoustic and combustion instability. PhD thesis, University of Cambridge (2015)

  95. Blumenthal, R.S., Tangirala, A.K., Sujith, R., Polifke, W.: A systems perspective on non-normality in low-order thermoacoustic models: full norms, semi-norms and transient growth. Int. J. Spray Combust. Dyn. 9(1), 19–43 (2016). https://doi.org/10.1177/1756827716652474

    Article  Google Scholar 

  96. Magri, L., Juniper, M.P., Moeck, J.P.: Sensitivity of the Rayleigh criterion in thermoacoustics. J. Fluid Mech. 882, R1 (2020). https://doi.org/10.1017/jfm.2019.860

    Article  MathSciNet  MATH  Google Scholar 

  97. Subramanian, P., Mariappan, S., Sujith, R.I., Wahi, P.: Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube. Int. J. Spray Combust. Dyn. 2(4), 325–355 (2011). https://doi.org/10.1260/1756-8277.2.4.325

    Article  Google Scholar 

  98. Subramanian, P., Sujith, R.I., Wahi, P.: Subcritical bifurcation and bistability in thermoacoustic systems. J. Fluid Mech. 715, 210–238 (2013). https://doi.org/10.1017/jfm.2012.514

    Article  MathSciNet  MATH  Google Scholar 

  99. Hermeth, S., Staffelbach, G., Gicquel, L.Y., Anisimov, V., Cirigliano, C., Poinsot, T.: Bistable swirled flames and influence on flame transfer functions. Combust. Flame 161(1), 184–196 (2014). https://doi.org/10.1016/j.combustflame.2013.07.022

    Article  Google Scholar 

Download references

Acknowledgements

F. Huhn is supported by Fundação para a Ciência e Tecnologia in Portugal under the Research Studentship No. SFRH/BD/134617/2017. L. Magri gratefully acknowledges financial support from the Royal Academy of Engineering Research Fellowships. Fruitful discussions with Prof. Q. Wang, Dr. P. Blonigan, N. Chandramoorthy and A. Ni are gratefully acknowledged. We are grateful to Dr. Patrick Blonigan for his insightful suggestions on using the filtering parameter.

Author information

Authors and Affiliations

  1. Cambridge University Engineering Department, Trumpington Street, Cambridge, CB2 1PZ, UK

    Francisco Huhn & Luca Magri

Authors
  1. Francisco Huhn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luca Magri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Magri.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Uncertainty on the estimation of infinitely time-averaged cost functionals

Uncertainty on the estimation of infinitely time-averaged cost functionals

We wish to estimate the value of an infinitely time-averaged cost functional, \(\langle \mathcal {J}\rangle \), from a collection of N independent samples of the time-averaged cost functional, \(\{\langle \mathcal {J}\rangle _T^{(i)}\}\), where the subscript T and superscript (i) represent the finite time used in the averaging and the index of the sample, respectively. From the Central Limit Theorem, we assume that the error in the estimation of \(\langle \mathcal {J}\rangle \) decays with the number of samples used to the power of \(-1/2\). Thus, we set a nonlinear constrained minimisation problem

$$\begin{aligned} \begin{aligned}&\underset{a, b}{\text {minimise}}&b \\&\text {subject to}&a - \frac{b}{\sqrt{n}}< \frac{1}{n}\sum _{i=1}^n \langle \mathcal {J}\rangle _T^{(i)} < a + \frac{b}{\sqrt{n}} \\&&\quad \quad \quad \quad \quad \quad \quad \; \forall \, n \in \{1, \cdots , N\} \end{aligned}. \end{aligned}$$
(36)

Thus, \(\langle \mathcal {J}\rangle \) should be in the range \(\big (a-b/\sqrt{N}, a+b/\sqrt{N}\big )\).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huhn, F., Magri, L. Optimisation of chaotically perturbed acoustic limit cycles. Nonlinear Dyn 100, 1641–1657 (2020). https://doi.org/10.1007/s11071-020-05582-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05582-x

Keywords

Search

Navigation