Skip to main content Accessibility help
Hostname: page-component-77ffc5d9c7-zwn4l Total loading time: 0.301 Render date: 2021-04-22T19:53:15.175Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Propagation speed of inertial waves in cylindrical swirling flows

Published online by Cambridge University Press:  19 September 2019

Alp Albayrak
Faculty of Mechanical Engineering, Technical University of Munich, D-85748 Garching, Germany
Matthew P. Juniper
Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
Wolfgang Polifke
Faculty of Mechanical Engineering, Technical University of Munich, D-85748 Garching, Germany
E-mail address:


Thermo-acoustic combustion instabilities arise from feedback between flow perturbations and the unsteady heat release rate of a flame in a combustion chamber. In the case of a premixed, swirl stabilized flame, an unsteady heat release rate results from acoustic velocity perturbations at the burner inlet on the one hand, and from azimuthal velocity perturbations, which are generated by acoustic waves propagating across the swirler, on the other. The respective time lags associated with these flow–flame interaction mechanisms determine the overall flame response to acoustic perturbations and therefore thermo-acoustic stability. The propagation of azimuthal velocity perturbations in a cylindrical duct is commonly assumed to be convective, which implies that the corresponding time lag is governed by the speed of convection. We scrutinize this assumption in the framework of small perturbation analysis and modal decomposition of the Euler equations by considering an initial value problem. The analysis reveals that azimuthal velocity perturbations in swirling flows should be regarded as dispersive inertial waves. As a result of the restoring Coriolis force, wave propagation speeds lie above and below the mean flow bulk velocity. The differences between wave propagation speed and convection speed increase with increasing swirl. A linear, time invariant step response solution for the dynamics of inertial waves is developed, which can be approximated by a concise analytical expression. This study enhances the understanding of the flame dynamics of swirl burners in particular, and contributes physical insight into the inertial wave dynamics in general.

JFM Papers
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.


Acharya, V. S. & Lieuwen, T. C. 2014 Role of azimuthal flow fluctuations on flow dynamics and global flame response of axisymmetric swirling flames. In 52nd Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics.Google Scholar
Albayrak, A., Bezgin, D. A. & Polifke, W. 2018a Response of a swirl flame to inertial waves. Intl J. Spray Combust. Dyn. 10 (4), 277286.CrossRefGoogle Scholar
Albayrak, A., Steinbacher, T., Komarek, T. & Polifke, W. 2018b Convective scaling of intrinsic thermo-acoustic eigenfrequencies of a premixed swirl combustor. Trans. ASME J. Engng Gas Turbines Power 140 (4), 041510.CrossRefGoogle Scholar
Arendt, S., Fritts, D. C. & Andreassen, Y. 1997 The initial value problem for Kelvin vortex waves. J. Fluid Mech. 344, 181212.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1999 Advanced Mathematical Methods for Scientists and Engineers. Springer.CrossRefGoogle Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14 (04), 593629.CrossRefGoogle Scholar
Blumenthal, R. S., Subramanian, P., Sujith, R. I. & Polifke, W. 2013 Novel perspectives on the dynamics of premixed flames. Combust. Flame 160 (7), 12151224.CrossRefGoogle Scholar
Candel, S., Durox, D., Schuller, T., Bourgouin, J. F. & Moeck, J. P. 2014 Dynamics of swirling flames. Annu. Rev. Fluid Mech. 46 (1), 147173.CrossRefGoogle Scholar
Culick, F. E. C. 1996 Combustion instabilities in propulsion systems. In Unsteady Combustion, 1st edn. (ed. Culick, F., Heitor, M. V. & Whitelaw, J. H.), NATO ASI Series, Series E: Applied Sciences, vol. 306, pp. 173241. Springer.CrossRefGoogle Scholar
Cumpsty, N. A. & Marble, F. E. 1977 The interaction of entropy fluctuations with turbine blade rows; a mechanism of turbojet engine noise. Proc. R. Soc. Lond. A 357 (1690), 323344.CrossRefGoogle Scholar
Fleifil, A., Annaswamy, A. M., Ghoneim, Z. A. & Ghoniem, A. F. 1996 Response of a laminar premixed flame to flow oscillations: a kinematic model and thermoacoustic instability results. Combust. Flame 106, 487510.CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J. M. 2003a Instability mechanisms in swirling flows. Phys. Fluids 15 (9), 26222639.CrossRefGoogle Scholar
Gallaire, F. & Chomaz, J. M. 2003b Mode selection in swirling jet experiments: a linear stability analysis. J. Fluid Mech. 494, 223253.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (2), 222224.CrossRefGoogle Scholar
Golubev, V. V. & Atassi, H. M. 1998 Acoustic–vorticity waves in swirling flows. J. Sound Vib. 209 (2), 203222.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. CUP Archive.Google Scholar
Greitzer, E. M., Tan, C. S. & Graf, M. B. 2004 Internal Flows. Cambridge University Press.CrossRefGoogle Scholar
Hirsch, C., Fanaca, D., Reddy, P., Polifke, W. & Sattelmayer, T. 2005 Influence of the swirler design on the flame transfer function of premixed flames. In ASME Turbo Expo 2005: Power for Land, Sea, and Air, pp. 151160. ASME.CrossRefGoogle Scholar
Juniper, M. P. 2012 Absolute and convective instability in gas turbine fuel injectors. In Volume 2: Combustion, Fuels and Emissions, Parts A and B, pp. 189198. ASME.CrossRefGoogle Scholar
Juniper, M. P., Hanifi, A. & Theofilis, V. 2014 Modal stability theory – lecture notes from the FLOW-NORDITA summer school on advanced instability methods for complex flows, Stockholm, Sweden, 2013. Appl. Mech. Rev. 66 (2), 024804.Google Scholar
Kaji, S. & Okazaki, T. 1970 Propagation of sound waves through a blade row: I. Analysis based on the semi-actuator disk theory. J. Sound Vib. 11 (3), 339353.Google Scholar
Kerrebrock, J. L. 1977 Small disturbances in turbomachine annuli with swirl. AIAA J. 15 (6), 794803.CrossRefGoogle Scholar
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81 (1), 206229.CrossRefGoogle Scholar
Komarek, T. & Polifke, W. 2010 Impact of swirl fluctuations on the flame response of a perfectly premixed swirl burner. J. Engng Gas Turbines Power 132 (6), 061503.CrossRefGoogle Scholar
Kousen, K. 1996 Pressure modes in ducted flows with swirl. In Aeroacoustics Conference. American Institute of Aeronautics and Astronautics.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63 (04), 753763.CrossRefGoogle Scholar
Munjal, M. L. 2014 Acoustics of Ducts and Mufflers, 2nd edn. Wiley.Google Scholar
Palies, P., Durox, D., Schuller, T. & Candel, S. 2010 The combined dynamics of swirler and turbulent premixed swirling flames. Combust. Flame 157 (9), 16981717.CrossRefGoogle Scholar
Palies, P., Durox, D., Schuller, T. & Candel, S. 2011a Acoustic–convective mode conversion in an aerofoil cascade. J. Fluid Mech. 672, 545569.CrossRefGoogle Scholar
Palies, P., Durox, D., Schuller, T. & Candel, S. 2011b Experimental study on the effect of swirler geometry and swirl number on flame describing functions. Combust. Sci. Technol. 183 (7), 704717.CrossRefGoogle Scholar
Palies, P., Ilak, M. & Cheng, R. 2017 Transient and limit cycle combustion dynamics analysis of turbulent premixed swirling flames. J. Fluid Mech. 830, 681707.Google Scholar
Palies, P., Schuller, T., Durox, D. & Candel, S. 2011c Modeling of premixed swirling flames transfer functions. Proc. Combust. Inst. 33 (2), 29672974.CrossRefGoogle Scholar
Parras, L. & Fernandez-Feria, R. 2007 Spatial stability and the onset of absolute instability of Batchelor’s vortex for high swirl numbers. J. Fluid Mech. 583, 2743.CrossRefGoogle Scholar
Poinsot, T. 2017 Prediction and control of combustion instabilites in real engines. Proc. Combust. Inst. 36, 128.CrossRefGoogle Scholar
Rayleigh, J. W. S. 1878 The explanation of certain acoustical phenomena. Nature 18, 319321.CrossRefGoogle Scholar
Renac, F., Sipp, D. & Jacquin, L. 2007 Criticality of compressible rotating flows. Phys. Fluids 19 (1), 018101.CrossRefGoogle Scholar
Richards, G. A. & Yip, M. J. 1995 Oscillating combustion from a premix fuel nozzle. In Comb. Inst./American Flame Research Committe Meeting, San Antonio, TX. USDOE Morgantown Energy Technology Center.Google Scholar
Rienstra, S. W. & Hirschberg, A.2018 An introduction to acoustics. Tech. Rep. IWDE 92-06. Eindhoven University of Technology.Google Scholar
Rusak, Z. & Lee, J. H. 2002 The effect of compressibility on the critical swirl of vortex flows in a pipe. J. Fluid Mech. 461, 301319.CrossRefGoogle Scholar
Saffman, P. G. 1993 Vortex Dynamics. Cambridge University Press.CrossRefGoogle Scholar
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 (1, 2), 2134.CrossRefGoogle Scholar
Steinbacher, T., Albayrak, A., Ghani, A. & Polifke, W. 2019 Response of premixed flames to irrotational and vortical velocity fields generated by acoustic perturbations. Proc. Combust. Inst. 37 (4), 53675375.CrossRefGoogle Scholar
Straub, D. L. & Richards, G. A. 1999 Effect of axial swirl vane location on combustion dynamics. In Volume 2: Coal, Biomass and Alternative Fuels; Combustion and Fuels; Oil and Gas Applications; Cycle Innovations, V002T02A014. ASME.Google Scholar
Tam, C. K. W. & Auriault, L. 1998 The wave modes in ducted swirling flows. J. Fluid Mech. 371, 120.Google Scholar
Thomson, W. 1880 XXIV. Vibrations of a columnar vortex. Phil. Mag. 10 (61), 155168.CrossRefGoogle Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 36
Total number of PDF views: 244 *
View data table for this chart

* Views captured on Cambridge Core between 19th September 2019 - 22nd April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Propagation speed of inertial waves in cylindrical swirling flows
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Propagation speed of inertial waves in cylindrical swirling flows
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Propagation speed of inertial waves in cylindrical swirling flows
Available formats

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *