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Continuous spectra of the Batchelor vortex

Published online by Cambridge University Press:  31 May 2011

XUERUI MAO
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
SPENCER SHERWIN*
Affiliation:
Department of Aeronautics, Imperial College London, South Kensington, London SW7 2AZ, UK
*
Email address for correspondence: s.sherwin@imperial.ac.uk

Abstract

The spectra of the Batchelor vortex are obtained by discretizing its linearized evolution operator using a modified Chebyshev polynomial approximation at a Reynolds number of 1000 and zero azimuthal wavenumber. Three types of eigenmodes are identified from the spectra: discrete modes, potential modes and free-stream modes. The discrete modes have been extensively documented but the last two modes have received little attention. A convergence study of the spectra and pseudospectra supports the classification that discrete modes correspond to discrete spectra while the other two modes correspond to continuous spectra. Free-stream modes have finite amplitude in the far field whilst potential modes decay to zero in the far field. The free-stream modes are therefore a limiting form of the potential modes when the radial decay rate of velocity components reduces to zero. The radial form of the free-stream modes with axial and radial wavenumbers is investigated and the penetration of the free-stream mode into the vortex core highlights the possibility of interaction between the potential region and the vortex core. A wavepacket pseudomode study confirms the existence of continuous spectra and predicts the locations and radial wavenumbers of the eigenmodes. The pseudomodes corresponding to the potential modes are observed to be in the form of one or two wavepackets while the free-stream modes are not observed to be in the form of wavepackets.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Abid, M. 2008 Nonlinear mode selection in a model of trailing line vortices. J. Fluid Mech. 605, 1945.CrossRefGoogle Scholar
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645658.CrossRefGoogle Scholar
Bender, C. & Orszag, S. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw Hill.Google Scholar
Boyd, J. 2001 Chebyshev and Fourier Spectrum Methods. Dover.Google Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl numbers. J. Fluid Mech. 500, 239262.CrossRefGoogle Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235274.CrossRefGoogle Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 3354.CrossRefGoogle Scholar
Gustavsson, L. H. 1979 Initial-value problem for boundary layer flows. Phys. Fluids 22, 16021605.CrossRefGoogle Scholar
Heaton, C. J. 2007 Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex. J. Fluid Mech. 576, 325348.CrossRefGoogle Scholar
Heinrichs, W. 1989 Improved condition number for spectral methods. Math. Comput. 53, 103119.CrossRefGoogle Scholar
Heinrichs, W. 1991 A stabilized treatment of the biharmonic operator with spectral methods. SIAM J. Sci. Stat. Comput. 12, 11621172.CrossRefGoogle Scholar
Jordinson, R. 1970 Spectrum of eigenvalues of the Orr–Sommerfeld equation for Blasius flow. Phys. Fluids 14, 25352537.CrossRefGoogle Scholar
Joshi, S. S. 1996 A systems theory approach to the control of plane Poiseuille flow. PhD thesis, Department of Electrical Engineering, UCLA.Google Scholar
Khorrami, M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197212.CrossRefGoogle Scholar
Khorrami, M. R. 1992 Behavior of asymmetric unstable modes of a trailing line vortex near the upper neutral curve. Phys. Fluids A 4, 13101313.CrossRefGoogle Scholar
Le Dizés, S. & Fabre, D. 2010 Viscous ring modes in vortices with axial jet. Theor. Comput. Fluid Dyn. 24, 349361.CrossRefGoogle Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769779.CrossRefGoogle Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753763.CrossRefGoogle Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497520.CrossRefGoogle Scholar
Mao, X. 2010 Vortex instability and transient growth. PhD thesis, Imperial College London.Google Scholar
McKernan, J. 2006 Control of plane Poiseuille flow: a theoretical and computational investigation. PhD thesis, Department of Aerospace Sciences, School of Engineering, Cranfield University.Google Scholar
Obrist, D. & Schmid, P. 2008 Resonance in the cochlea with wave packet pseudomodes. In Proc. 22nd Intl Congress of Theoretical and Applied Mechanics. Adelaide, Australia.Google Scholar
Obrist, D. & Schmid, P. 2009 Wave packet pseudomodes upstream of a swept cylinder. In Seventh IUTAM Symposium on Laminar–Turbulent Transition. Stockholm, Sweden.Google Scholar
Obrist, D. & Schmid, P. J. 2003 a On the linear stability of swept attachment-line boundary layer flow. Part 1. Spectrum and asymptotic behaviour. J. Fluid Mech. 493, 129.CrossRefGoogle Scholar
Obrist, D. & Schmid, P. J. 2003 b On the linear stability of swept attachment-line boundary layer flow. Part 2. Non-modal effects and receptivity. J. Fluid Mech. 493, 3158.CrossRefGoogle Scholar
Obrist, D. & Schmid, P. J. 2010 Algebraically decaying modes and wave packet pseudo-modes in swept Hiemenz flow. J. Fluid Mech. 643, 309332.CrossRefGoogle Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148154.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Trefethen, L. N. 2005 Wave packet pseudomodes of variable coefficient differential operators. Proc. R. Soc. Lond. A 461, 30993122.Google Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465519.CrossRefGoogle Scholar
Zaki, T. A. & Saha, S. 2009 On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers. J. Fluid Mech. 626, 111147.CrossRefGoogle Scholar