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Global linear stability of the non-parallel Batchelor vortex

Published online by Cambridge University Press:  15 June 2009

C. J. HEATON*
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS–École Polytechnique, 91128 Palaiseau, France
J. W. NICHOLS
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS–École Polytechnique, 91128 Palaiseau, France
P. J. SCHMID
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS–École Polytechnique, 91128 Palaiseau, France
*
Email address for correspondence: cjh43@damtp.cam.ac.uk

Abstract

Linear stability of the non-parallel Batchelor vortex is studied using global modes. This family of swirling wakes and jets has been extensively studied under the parallel-flow approximation, and in this paper we extend to more realistic non-parallel base flows. Our base flow is obtained as an exact steady solution of the Navier–Stokes equations by direct numerical simulation (with imposed axisymmetry to damp all instabilities). Global stability modes are computed by numerical simulation of the linearized equations, using the implicitly restarted Arnoldi method, and we discuss fully the numerical and convergence issues encountered. Emphasis is placed on exploring the general structure of the global spectrum, and in particular the correspondence between global modes and local absolute modes which is anticipated by weakly non-parallel asymptotic theory. We believe that our computed global modes for a weakly non-parallel vortex are the first to display this correspondence with local absolute modes. Superpositions of global modes are also studied, allowing an investigation of the amplifier dynamics of this unstable flow. For an illustrative case we find global non-modal transient growth via a convective mechanism. Generally amplifier dynamics, via convective growth, are prevalent over short time intervals, and resonator dynamics, via global mode growth, become prevalent at later times.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

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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
Abid, M. & Brachet, M. E. 1998 Direct numerical simulations of the Batchelor trailing vortex by a spectral method. Phys. Fluids 10, 469475.CrossRefGoogle Scholar
Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. b/Fluids 27, 501513.CrossRefGoogle Scholar
Åkervik, E., Hoepffner, J., Ehrenstein, U. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579, 305314.CrossRefGoogle Scholar
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645658.CrossRefGoogle Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271304.CrossRefGoogle Scholar
Broadhurst, M. S. 2006 Vortex stability and breakdown: direct numerical simulation and stability analysis using biglobal and parabolised formulations. PhD thesis, Imperial College, London.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.CrossRefGoogle Scholar
Constantinescu, G. S. & Lele, S. K. 2002 A highly accurate technique for the treatment of flow equations at the polar axis in cylindrical coordinates using series expansions. J. Comput. Phys. 183, 165186.CrossRefGoogle Scholar
Cossu, C. & Chomaz, J. M. 1997 Global measures of local convective instabilities. Phys. Rev. Lett. 78, 43874390.CrossRefGoogle Scholar
Delbende, I., Chomaz, J. M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229254.CrossRefGoogle Scholar
Doaré, O. & De Langre, E. 2006 The role of boundary conditions in the instability of one-dimensional systems. Eur. J. Mech. b/Fluids 25, 948959.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209218.CrossRefGoogle Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl number. J. Fluid Mech. 500, 239262.CrossRefGoogle Scholar
Griffith, M. D., Leweke, T., Thompson, M. C. & Hourigan, K. 2008 Steady inlet flow in stenotic geometries: convective and absolute instabilities. J. Fluid Mech. 616, 111133.CrossRefGoogle Scholar
Heaton, C. J. 2007 a Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex. J. Fluid Mech. 576, 325348.CrossRefGoogle Scholar
Heaton, C. J. 2007 b Optimal growth of the Batchelor vortex viscous modes. J. Fluid Mech. 592, 495505.CrossRefGoogle Scholar
Heaton, C. J. & Peake, N. 2007 Transient growth in vortices with axial flow. J. Fluid Mech. 587, 271301.CrossRefGoogle Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29, 245283.CrossRefGoogle Scholar
Huerre, P. 2000 Open shear flow instabilities. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 159230. Cambridge University Press.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed. Godreche, C. & Manneville, P.), pp. 81294. Cambridge University Press.CrossRefGoogle Scholar
Khorrami, M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197212.CrossRefGoogle Scholar
Le Dizès, S. & Fabre, D. 2007 Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices. J. Fluid Mech. 585, 153180.CrossRefGoogle Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1998 ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM.CrossRefGoogle Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Annu. Rev. Fluid Mech. 10, 221246.CrossRefGoogle Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335356.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
Marquet, O., Lombardi, M., Chomaz, J. M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 121.CrossRefGoogle Scholar
Marquet, O., Sipp, D., Chomaz, J. M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429443.CrossRefGoogle Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91114.CrossRefGoogle Scholar
Nichols, J. W. 2005 Simulation and stability analysis of jet diffusion flames. PhD thesis, University of Washington.Google Scholar
Nichols, J. W., Chomaz, J. M. & Schmid, P. J. 2009 Twisted absolute instability in lifted flames. Phys. Fluids 21, 015110.CrossRefGoogle Scholar
Nichols, J. W. & Schmid, P. J. 2008 The effect of a lifted flame on the stability of round fuel jets. J. Fluid Mech. 609, 275284.CrossRefGoogle Scholar
Nichols, J. W., Schmid, P. J. & Riley, J. J. 2007 Self-sustained oscillations in variable-density round jets. J. Fluid Mech. 582, 341376.CrossRefGoogle Scholar
Olendraru, C. & Sellier, A. 2002 Viscous effects in the absolute-convective instability of the Batchelor vortex. J. Fluid Mech. 459, 371396.CrossRefGoogle Scholar
Olendraru, C., Sellier, A., Rossi, M. & Huerre, P. 1999 Inviscid instability of the Batchelor vortex: absolute-convective transition and spatial branches. Phys. Fluids 11, 18051820.CrossRefGoogle Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407417.CrossRefGoogle Scholar
Pier, B. 2008 Local and global instabilities in the wake of a sphere. J. Fluid Mech. 603, 3961.CrossRefGoogle Scholar
Ruith, M. R., Chen, P., Meiburg, E. & Maxworthy, T. 2003 Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486, 331378.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Stewartson, K. & Brown, S. N. 1985 Near-neutral centre-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387399.CrossRefGoogle Scholar
Yin, X., Sun, D. & Wei, M. 2000 Absolute and convective instability character of slender vortices. Phys. Fluids 12, 10621072.CrossRefGoogle Scholar