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Linearized oscillations of a vortex column: the singular eigenfunctions

Published online by Cambridge University Press:  20 February 2014

Anubhab Roy Open the ORCID record for Anubhab Roy [Opens in a new window]  and
Ganesh Subramanian
Anubhab Roy
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560 064, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560 064, India
*
†Email address for correspondence: [email protected]
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Abstract

In 1880 Lord Kelvin analysed the linearized inviscid oscillations of a Rankine vortex as part of a theory of vortex atoms. These eponymously named neutrally stable modes are, however, exceptional regular oscillations that make up the discrete spectrum of the Rankine vortex. In this paper, we examine the singular oscillations that make up the continuous spectrum (CS) and span the entire base state range of frequencies. In two dimensions, the CS eigenfunctions have a twin-vortex-sheet structure similar to that known from earlier investigations of parallel flows with piecewise linear velocity profiles. The vortex sheets are cylindrical, being threaded by axial lines, with one sheet at the edge of the core and the other at the critical radius in the irrotational exterior; the latter refers to the radial location at which the fluid co-rotates with the eigenmode. In three dimensions, the CS eigenfunctions have core vorticity and may be classified into two families based on the singularity at the critical radius. For the first family, the singularity is a cylindrical vortex sheet threaded by helical vortex lines, while for the second family it has a localized dipole structure with radial vorticity. The presence of perturbation vorticity in the otherwise irrotational exterior implies that the CS modes, unlike the Kelvin modes, offer a modal interpretation for the (linearized) interaction of the Rankine vortex with an external vortical disturbance. It is shown that an arbitrary initial distribution of perturbation vorticity, both in two and three dimensions, may be evolved as a superposition over the discrete and CS modes; this modal representation being equivalent to a solution of the corresponding initial value problem. For the restricted case of an initial axial vorticity distribution in two dimensions, the modal representation may be generalized to a smooth vortex. Finally, for the three-dimensional case, the analogy between rotational flows and stratified shear flows, and the known analytical solution for stratified Couette flow, are used to clarify the singular manner in which the modal superposition for a smooth vortex approaches the Rankine limit.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex instability Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 741 , 25 February 2014 , pp. 404 - 460
Copyright
© 2014 Cambridge University Press 

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References

Antkowiak, A. & Brancher, P. 2004 Transient growth for the Lamb–Oseen vortex. Phys. Fluids 16, L1–L4.Google Scholar
Arendt, S., Fritts, D. & Andreassen, Ø. 1997 The initial value problem for Kelvin vortex waves. J. Fluid Mech. 344, 181–212.Google Scholar
Balmforth, N. J. & Morrison, P. J. 1995 Singular eigenfunctions for shearing fluids I. Institute for Fusion studies, University of Texas, Austin, Report No. 692.Google Scholar
Balmforth, N. J., Smith, S. G. L. & Young, W. R. 2001 Disturbing vortices. J. Fluid Mech. 426, 95–133.Google Scholar
Banks, W. H. H., Drazin, P. G. & Zaturska, M. B. 1976 On the normal modes of parallel flow of inviscid stratified fluid. J. Fluid Mech. 75, 149–271.Google Scholar
Bassom, A. P. & Gilbert, A. D. 1998 The spiral wind-up of vorticity in an inviscid planar vortex. J. Fluid Mech. 371, 109–140.Google Scholar
Batchelor, G. K. 1967 Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Benney, D. J. & Bergeron, R. F. 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48, 181–204.Google Scholar
Booker, J. R. & Bretherton, F. P. 1967 The critical layer for internal gravity waves in a shear flow. J. Fluid Mech. 27, 513–539.Google Scholar
Bretherton, F. P. 1967 The time-dependent motion due to a cylinder moving in an unbounded rotating or stratified fluid. J. Fluid Mech. 28, 545–570.Google Scholar
Briggs, R. J., Daugherty, J. D. & Levy, R. H. 1970 Role of Landau damping in cross-field electron beams and inviscid shear flow. Phys. Fluids 13, 421–432.CrossRefGoogle Scholar
Brown, S. N. & Stewartson, K. D. 1980 On the algebraic decay of disturbances in a stratified linear shear flow. J. Fluid Mech. 100, 811–816.CrossRefGoogle Scholar
Case, K. M. 1959 Plasma oscillations. Ann. Phys. (NY) 7, 349–364.CrossRefGoogle Scholar
Case, K. M. 1960a Stability of inviscid plane Couette flow. Phys. Fluids 3, 143–148.Google Scholar
Case, K. M. 1960b Stability of idealized atmosphere. I. Discussion of results. Phys. Fluids 3, 149–154.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Davis, R. E. 1969 On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36, 337–346.Google Scholar
Dikii, L. A. 1960 The stability of plane-parallel flows of an ideal fluid. Sov. Phys. Dokl. 135, 1179–1182.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dyson, F. J. 1960 Stability of idealized atmosphere. II. Zeros of the confluent hypergeometric function. Phys. Fluids 3, 155–157.Google Scholar
Eliassen, A., Hoiland, E. & Riis, E. 1953 Two-Dimensional Perturbation of a Flow with Constant Shear of a Stratified Fluid. Inst. Weather and Climate Res., publ. no. 1.Google Scholar
Engevik, L. 1971 A note on a stability problem in hydrodynamics. Acta Mech. 12, 143–153.Google Scholar
Fabre, D. 2002 Instabilités et instationnarités dans les tourbillons: application aux sillages avions. PhD thesis, Université Paris VI.Google Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortex at large swirl number. J. Fluid Mech. 500, 239–262.Google Scholar
Fabre, D. & Le Dizès, S. 2008 Viscous and inviscid centre modes in the linear stability of vortices: the vicinity of the neutral curves. J. Fluid Mech. 603, 1–38.Google Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 Kelvin waves and the singular modes of the Lamb Oseen vortex. J. Fluid Mech. 551, 235–274.Google Scholar
Fadeev, L. D. 1971 On the theory of the stability of stationary plane-parallel flows of an ideal fluid. Zapiski Nauch. Semin. Leningrad. Otdel. Matemat. Instit. Akad. Nauk SSSR 21, 164–172.Google Scholar
Farrell, B. F. 1984 Modal and non-modal baroclinic waves. J. Atmos. Sci. 41, 668–673.Google Scholar
Farrell, B. F. 1989 Optimal excitation of baroclinic waves. J. Atmos. Sci. 46, 1193–1206.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993b Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5, 2600–2609.Google Scholar
Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. 316, 29–62.Google Scholar
Friedman, B. 1990 Principles and Techniques of Applied Mathematics. Dover.Google Scholar
Gakhov, F. D. 1990 Boundary Value Prolems. Dover.Google Scholar
Gel’fand, I. M. & Shilov, G. E. 1964 Generalized Functions, Vol. 1 – Properties and Operations. Academic.Google Scholar
Graham, M. D. 1998 Effect of axial flow on viscoelastic Taylor–Couette instability. J. Fluid Mech. 360, 341–374.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Heaton, C. J. 2007a Centre modes in inviscid vortex flows, and their application to the stability of the Batchelor vortex. J. Fluid Mech. 576, 325–348.CrossRefGoogle Scholar
Heaton, C. J. 2007b Optimal growth of the Batchelor vortex viscous modes. J. Fluid Mech. 592, 495–505.Google Scholar
Heaton, C. J. & Peake, N. 2006 Algebraic and exponential instability of inviscid swirling flow. J. Fluid Mech. 565, 279–318.Google Scholar
Heaton, C. J. & Peake, N. 2007 Transient growth in vortices with axial flow. J. Fluid Mech. 587, 271–301.Google Scholar
Held, I. M. 1985 Pseudomomentum and the orthogonality of modes in shear flows. J. Atmos. Sci. 42, 2280–2288.Google Scholar
Hirota, M., Tatsuno, T. & Yoshida, Z. 2003 Degenerate continuous spectra producing localized secular instability – an example in a non-neutral plasma. J. Plasma Phys. 69, 397–412.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Ince, E. L. 1956 Ordinary Differential Equations. Dover.Google Scholar
Kelbert, M. & Sazonov, I. 1996 Pulses and other Wave Processes in Fluids. Kluwer.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155–168.Google Scholar
Khorrami, M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197–212.Google Scholar
Kopiev, V. F. & Chernyshev, S. A. 1997 Vortex ring eigen-oscillations as a source of sound. J. Fluid Mech. 341, 19–57.CrossRefGoogle Scholar
Kupferman, R. 2005 On the linear stability of plane Couette flow for an Oldroyd-B fluid and its numerical approximation. J. Non-Newton. Fluid Mech. 127, 169–190.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.Google Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.Google Scholar
Le Dizès, S. 2000 Non-axisymmetric vortices in two-dimensional flows. J. Fluid Mech. 406, 175–198.Google Scholar
Le Dizès, S. 2004 Viscous critical-layer analysis of vortex normal modes. Stud. Appl. Math. 112, 315–332.Google Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.Google Scholar
Le Dizès, S. & Fabre, D. 2007 Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices. J. Fluid Mech. 585, 153–180.Google Scholar
Le Dizès, S. & Lacaze, L. 2005 Non-axisymmetric vortices in two-dimensional flows. J. Fluid Mech. 542, 69–96.Google Scholar
Leibovich, S., Brown, S. N. & Patel, Y. 1986 Bending waves on inviscid columnar vortices. J. Fluid Mech. 173, 595–624.Google Scholar
Leibovich, S. & Ma, H. Y. 1983 Soliton propagation on vortex cores and the Hasimoto soliton. Phys. Fluids 26, 3173–3179.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part I. Inviscid theory. J. Fluid Mech. 63, 753–763.Google Scholar
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Maslowe, S. 1986 Critical layers in shear flows. Annu. Rev. Fluid Mech. 18, 405–432.Google Scholar
Maslowe, S. & Nigam, N. 2008 The nonlinear critical layer for Kelvin modes on a vortex with a continuous velocity profile. SIAM J. Appl. Math. 68, 825–843.Google Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.Google Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.Google Scholar
Melander, M. V. & Hussain, F. 1993 Coupling between a coherent structure and fine-scale turbulence. Phys. Rev. E 48, 2669–2689.Google Scholar
Melander, M. V. & Hussain, F. 1994 Core dynamics on a vortex column. Fluid Dyn. Res. 13, 1–37.Google Scholar
Michalke, A. & Timme, A. 1967 On the inviscid instability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, 647–666.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 402, 349–378.Google Scholar
Orr, W. McF. 1907 Stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: A perfect liquid. Proc. R. Irish. Acad. A 27, 9–68.Google Scholar
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251–288.Google Scholar
Pradeep, D. S. & Hussain, F. 2010 Vortex dynamics of turbulence-coherent structure interaction. Theor. Comput. Fluid Dyn. 24, 265–282.Google Scholar
Rallison, J. M. & Hinch, E. J. 1995 Instability of a high-speed submerged elastic jet. J. Fluid Mech. 288, 311–324.Google Scholar
Roy, A. 2013 Singular eigenfunctions in hydrodynamic stability: the roles of rotation, stratification and elasticity. PhD thesis, Jawaharlal Nehru Centre for Advanced Scientific Research.Google Scholar
Roy, A. & Subramanian, G. 2012 Normal mode interpretation of ‘lift-up’ effect. J. Fluid Mech. (submitted).Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sazonov, I. A. 1989 Interaction of continuous spectrum waves with each other and discrete spectrum waves. Fluid Dyn. Res. 4, 586–592.Google Scholar
Sazonov, I. A. 1996 Evolution of three-dimensional wave packets in the Couette flow. Izv. Atmos. Ocean. Phys. 32, 21–28.Google Scholar
Schecter, D. A., Durbin, D. H. D., Cass, A. C., Dritscoll, C. F., Lansky, I. M. & O’Neil, T. M. 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12, 2397–2412.Google Scholar
Schecter, D. A. & Montgomery, M. T. 2003 On the symmetrization rate of an intense geophysical vortex. Dyn. Atmos. Oceans 37, 55–88.CrossRefGoogle Scholar
Schimd, P. J. & Henningson, D. S. 2001 Stability and Transition in Fluid Flows. Springer.Google Scholar
Shore, S. N. 1992 An Introduction to Astrophysical Hydrodynamics. Academic.Google Scholar
Shrira, V. I. & Sazonov, I. A. 2001 Quasi-modes in boundary-layer-type flows. Part 1. Inviscid two-dimensional spatially harmonic perturbations. J. Fluid Mech. 446, 133–171.Google Scholar
Shrira, V. I. & Sazonov, I. A. 2003 Quasi-modes in boundary-layer-type flows. Part 2. Large-time asymptotics of broadband inviscid small-amplitude two-dimensional perturbations. J. Fluid Mech. 488, 245–282.Google Scholar
Spalart, P. R. 1998 Airplane trailing vortices. Annu. Rev. Fluid Mech. 30, 107–138.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid modes with critical layers. IMA J. Appl. Maths 27, 133–176.Google Scholar
Stewartson, K., Ng, T. W. & Brown, S. 1988 Viscous centre modes in the stability of swirling Poiseuille flow. Phil. Trans. R. Soc. Lond. A 324, 473–512.Google Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. A 132, 499–523.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–583.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Van Kampen, G. 1955 On the theory of stationary waves in plasmas. Physica 51, 949–963.Google Scholar
Watson, G. N. 1927 Theory of Bessel Functions. Cambridge University Press.Google Scholar
Widnall, S. 1975 The structure and dynamics of vortex filaments. Annu. Rev. Fluid Mech. 7, 141–165.Google Scholar
Yih, C. S. 1980 Stratified Flows. Academic.Google Scholar