Journal of Engineering Mathematics

, Volume 74, Issue 1, pp 37–52 | Cite as

Impact of curvature on the kinematic response of small flames

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Abstract

The objective of the study is to investigate the impact of curvature on the kinematic response of an axisymmetric curved laminar premixed flame, utilizing a phenomenological relationship between the curvature and the laminar burning velocity. The reference and the perturbed flame shapes are obtained by numerically or analytically integrating the flame kinematics equations. The steady reference flame shape deviates gradually from that obtained without the curvature effect as the effect of curvature intensifies. By having its tip rounded off, the overall flame height under curvature becomes lower. Perturbed flame shapes with curvature show more attenuated undulation than those without curvature effect. Linear perturbation analysis is performed to determine the frequency-domain response of the instantaneous heat release rate to upstream flow perturbations. In the low frequency range, results are qualitatively similar to those obtained using those obtained using the model without curvature, but substantial differences in the phase shift are observed as the frequency of perturbation increases. Flames under weak curvature effects show behaviors similar to a first-order filter, whose cutoff frequency increases as the effect of curvature becomes stronger. Careful comparison reveals that the flame is mainly influenced by the modification of the burning velocity due to the curvature of the mean reference flame at the low frequency range while the influence of the time-varying perturbed burning velocity increases as the frequency of perturbation increases. A rule for scaling the response is proposed to match the results with and without the curvature effect, based on asymptotic analysis of the flame kinematics. Rescaling is effective for the low-frequency range, and a good correspondence between the results with and without the curvature effect is achieved.

Keywords

Conical flames Curvature Flame kinematics Transfer function Thermoacoustic instability 

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References

  1. 1.
    Rayleigh JWS (1878) The explanation of certain acoustic phenomena. Nature 18: 319–321ADSCrossRefGoogle Scholar
  2. 2.
    Dowling AP (2000) Instability in lean premixed combustors. Proc Inst Mech Eng A J Power Energy 214: 317–331CrossRefGoogle Scholar
  3. 3.
    Ducruix S, Schuller T, Durox D, Candel S (2003) Combustion dynamics and instabilities: elementary coupling and driving mechanisms. J Propuls Power 19: 722–734CrossRefGoogle Scholar
  4. 4.
    Lieuwen T (2003) Modeling premixed combustion-acoustic wave interactions: a review. J Propuls Power 19: 765–781CrossRefGoogle Scholar
  5. 5.
    Fleifil MF, Annaswamy AM, Ghoneim ZA, Ghoniem AF (1996) Response of a laminar premixed flame to flow oscillations: a kinematic model and thermoacoustic instability results. Combust Flame 106: 487–510CrossRefGoogle Scholar
  6. 6.
    Dowling AP (1999) A kinematic model of a ducted flame. J Fluid Mech 394: 51–72ADSMATHCrossRefGoogle Scholar
  7. 7.
    Mehta PG, Soteriou MC, Banaszuk A (2005) Impact of exothermicity on steady and linearized response of a premixed ducted flame. Combust Flame 141: 392–405CrossRefGoogle Scholar
  8. 8.
    Ducruix S, Durox D, Candel S (2000) Theoretical and experimental determinations of the transfer function of a laminar premixed flame. Proc Combust Inst 28: 765–773CrossRefGoogle Scholar
  9. 9.
    Rook R, de Goey LPH, Somers LMT, Schreel KRAM, Parchen R (2002) Response of burner-stabilized flat flames to acoustic perturbations. Combust Theory Model 6: 223–242ADSCrossRefGoogle Scholar
  10. 10.
    Altay HM, Park S, Wu D, Wee D, Annaswamy AM, Ghoniem AF (2009) Modeling the dynamic response of a laminar perforated-plate stabilized flame. Proc Combust Inst 32: 1359–1366CrossRefGoogle Scholar
  11. 11.
    Markstein GH (1951) Experimental and theoretical studies of flame-front stability. J Aeronaut Sci 18: 199–209Google Scholar
  12. 12.
    Gu XJ, Haq MZ, Lawes M, Woolley R (2000) Laminar burning velocity and Markstein lengths of methane-air mixtures. Combust Flame 124: 41–58CrossRefGoogle Scholar
  13. 13.
    Eckhaus W (1961) Theory of flame-front stability. J Fluid Mech 10: 80–100MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Joulin G (1994) On the response of premixed flames to time-dependent stretch and curvature. Combust Sci Technol 97: 219–229CrossRefGoogle Scholar
  15. 15.
    Law CK, Sung CJ (2000) Structure, aerodynamics, and geometry of premixed flamelets. Prog Energy Combust Sci 26: 459–505CrossRefGoogle Scholar
  16. 16.
    Noiray N, Durox D, Schuller T, Candel S (2007) Passive control of combustion instabilities involving premixed flames anchored on perforated plates. Proc Combust Inst 31: 1283–1290CrossRefGoogle Scholar
  17. 17.
    Durox D, Schuller T, Noiray N, Candel S (2009) Experimental analysis of nonlinear flame transfer functions for different flame geometries. Proc Combust Inst 32: 1391–1398CrossRefGoogle Scholar
  18. 18.
    Gray A (1993) Modern differential geometry of curves and surfaces. CRC Press, Boca RatonMATHGoogle Scholar
  19. 19.
    Van Dyke M (1975) Perturbation methods in fluid mechanics. Parabolic Press, StanfordMATHGoogle Scholar
  20. 20.
    Shampine LF, Gladwell I, Thompson S (2003) Solving ODEs with MATLAB. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  21. 21.
    McOwen RC (1996) Partial differential equations: methods and applications. Prentice Hall, Upper Saddle RiverMATHGoogle Scholar
  22. 22.
    Schuller T, Ducruix S, Durox D, Candel S (2002) Modeling tools for the prediction of premixed flame transfer functions. Proc Combust Inst 29: 107–113CrossRefGoogle Scholar
  23. 23.
    Schuller T, Durox D, Candel S (2003) A unified model for the prediction of laminar flame transfer functions: comparisons between conical and V-flame dynamics. Combust Flame 134: 21–34CrossRefGoogle Scholar
  24. 24.
    Lee DH, Lieuwen T (2003) Premixed flame kinematics in a longitudinal acoustic field. J Propuls Power 19: 837–846CrossRefGoogle Scholar
  25. 25.
    Wang HY, Law CK, Lieuwen T (2009) Linear response of stretch-affected premixed flames to flow oscillations. Combust Flame 156: 889–895CrossRefGoogle Scholar
  26. 26.
    Preetham, Thumuluru SK, Hemchandra S, Lieuwen T (2010) Linear response of laminar premixed flames to flow oscillations: unsteady stretch effects. J Propuls Power 26: 524–532CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Environmental Science and EngineeringEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Bosch ThermotechnologyDeventerThe Netherlands
  3. 3.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

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