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The preferred mode of incompressible jets: linear frequency response analysis

  • X. Garnaud (a1), L. Lesshafft (a1), P. J. Schmid (a1) and P. Huerre (a1)


The linear amplification of axisymmetric external forcing in incompressible jet flows is investigated within a fully non-parallel framework. Experimental and numerical studies have shown that isothermal jets preferably amplify external perturbations for Strouhal numbers in the range $0. 25\leq {\mathit{St}}_{D} \leq 0. 5$ , depending on the operating conditions. In the present study, the optimal forcing of an incompressible jet is computed as a function of the excitation frequency. This analysis characterizes the preferred amplification as a pseudo-resonance with a dominant Strouhal number of around $0. 45$ . The flow response at this frequency takes the form of a vortical wavepacket that peaks inside the potential core. Its global structure is characterized by the cooperation of local shear-layer and jet-column modes.


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Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134.
Balay, S., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F. & Zhang, H. 2008 PETSc users’ manual. Tech. Rep. ANL-95/11, Revision 3.0.0. Argonne National Laboratory, available at
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750756.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14 (4), 529.
Cooper, A. J. & Crighton, D. G. 2000 Global modes and superdirective acoustic radiation in low-speed axisymmetric jets. Eur. J. Mech. B 19 (5), 559574.
Crighton, D. G. & Gaster, M. 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77 (2), 397.
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547.
Dick, E. 2009 Introduction to finite element methods in computational fluid dynamics. In Computational Fluid Dynamics: An Introduction, 3rd edn. Springer.
Garnaud, X. 2012 Modes, transient dynamics and forced response of circular jets. PhD thesis, Ecole Polytechnique.
Garnaud, X., Lesshafft, L. & Huerre, P. 2011 Global linear stability of a model subsonic jet. AIAA Paper 2011-3608.
Gudmundsson, K. & Colonius, T. 2011 Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97128.
Hecht, F. 2011 FreeFem++ manual, 3rd edn, version 3.16-1. Tech. Rep. Available at
Hernandez, V., Roman, J. E. & Vidal, V. 2005 SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31 (3), 351362.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Jendoubi, S. & Strykowski, P. J. 1994 Absolute and convective instability of axisymmetric jets with external flow. Phys. Fluids 6 (9), 3000.
Lesshafft, L. 2007 Global modes and aerodynamic sound radiation in self-excited hot jets. PhD thesis, Ecole Polytechnique.
Marquet, O. & Sipp, D. 2010 Global sustained perturbations in a backward-facing step flow. In Seventh IUTAM Symposium on Laminar–Turbulent Transition. Springer.
Matsushima, T. & Marcus, P. S. 1995 A spectral method for polar coordinates. J. Comput. Phys. 120, 365374.
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159199.
Monkewitz, P. A. 1989 Feedback control of global oscillations in fluid systems. AIAA Paper 89-0991.
Monkewitz, P. A. & Sohn, K. 1988 Absolute instability in hot jets. AIAA J. 26 (8), 911916.
Monokrousos, A., Akervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181.
Moore, C. J. 1977 The role of shear-layer instability waves in jet exhaust noise. J. Fluid Mech. 80 (2), 321.
Nichols, J. W. & Lele, S. K. 2010 Global mode analysis of turbulent high-speed jets. Annual Research Briefs 2010, Center for Turbulence Research.
Nichols, J. W. & Lele, S. K. 2011a Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225241.
Nichols, J. W. & Lele, S. K. 2011b Non-normal global modes of high-speed jets. Intl J. Spray Combust. Dyn. 3 (4), 285302.
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458.
Ray, P. K., Cheung, L. C. & Lele, S. K. 2009 On the growth and propagation of linear instability waves in compressible turbulent jets. Phys. Fluids 21 (5), 054106.
Rodriguez Alvarez, D., Samanta, A., Cavalieri, A. V. G., Colonius, T. & Jordan, P. 2011 Parabolized stability equation models for predicting large-scale mixing noise of turbulent round jets. In Proceedings of the 17th AIAA/CEAS Aeroacoustics Conference, Portland, Oregon. AIAA Paper 2011-2743.
Sipp, D. & Marquet, O. 2012 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn., doi:10.1007/s00162-012-0265-y.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578584.
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