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Transient growth in vortices with axial flow

Published online by Cambridge University Press:  31 August 2007

C. J. HEATON  and
N. PEAKE
C. J. HEATON
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
N. PEAKE
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
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Abstract

We investigate transient growth in high-Reynolds-number vortices with axial flow. Manycases of vortex instability are not fully explained by strong exponential instability modes, and transient growth could offer an alternative route to breakdown in such cases. Strong transient growth is found, in agreement with previous studies. We first discuss the problem by reference to ducted vortices which model aeroengine flow. The transient growth is inviscid in character, and in this paper we specifically interpret it as an effect of the inviscid continuous spectrum. The relevant inviscid theory explains new scalings which we find for the transient growth, which are generalizations of the quadratic scaling seen previously in two-dimensional flows and non-swirling pipe flows. We then turn to a second case, of interest for vortex breakdown, the Batchelor vortex, and present calculations of the transient growth. Large growth is possible, especially for the helical modes (with azimuthal wavenumber |m| = 1). The general trends are complicated by a number ofissues, including a long-wavelength effect and a resonance effect, both of which were recently discovered for a vortex without axial flow and are found here to be present in the Batchelor vortex also. Overall, the results suggest that strong transient effects are present in the moderate- to high-swirl regime of practical interest (swirl number q ≳ 2). Foraxisymmetric (m = 0) and higher (|m| > 1) modes, however, transient effects are not found to be significant.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 587 , 25 September 2007 , pp. 271 - 301
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Antkowiak, A. & Brancher, P. 2004 Transient growth for the Lamb–Oseen vortex. Phys. Fluids. 16, L1–L4.CrossRefGoogle Scholar
Atassi, H. M., Ali, A. A., Atassi, O. V. & Vinogradov, I. V. 2004 Scattering of incident disturbances by an annular cascade in a swirling flow. J. Fluid Mech. 499, 111–138.Google Scholar
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.CrossRefGoogle Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.CrossRefGoogle Scholar
Ben-Dov, G., Levinski, V. & Cohen, J. 2004 Optimal disturbances in swirling flows. AIAA J. 42, 1841–1848.CrossRefGoogle Scholar
Bernoff, A. J. & Lingevitch, J. F. 1994 Rapid relaxation of an axisymmetric vortex. Phys. Fluids. 6, 3717–3723.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods 2nd edn. Dover.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids. 4, 1637–1650.CrossRefGoogle Scholar
Chapman, S. J. 2002 Subcritical transition in channel flows. J. Fluid Mech. 451, 35–97.CrossRefGoogle Scholar
Cooper, A. J. & Peake, N. 2005 Upstream-radiated rotor–stator interaction noise in mean swirling flow. J. Fluid Mech. 523, 219–250.CrossRefGoogle Scholar
Cooper, A. J. & Peake, N. 2006 Transient growth and rotor–stator interaction noise in mean swirling duct flow. J. Sound Vib. 295, 553–570.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids. 12, 120–130.CrossRefGoogle Scholar
Duck, P. W. & Foster, M. R. 1980 The inviscid stability of a trailing line vortex. Z. Angew. Math. Phys. 31, 524–532.CrossRefGoogle Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl number. J. Fluid Mech. 500, 239–262.CrossRefGoogle 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
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5, 2600–2609.CrossRefGoogle Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.CrossRefGoogle Scholar
Hanifi, A. & Henningson, D. S. 1998 The compressible inviscid algebraic instability for streamwise independent disturbances. Phys. Fluids. 10, 1784–1786.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, 325–348.CrossRefGoogle Scholar
Heaton, C. J. & Peake, N. 2006 Algebraic and exponential instability of inviscid swirling flow. J. Fluid Mech. 565, 279–318.CrossRefGoogle Scholar
Khorrami, M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197–212.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, 206–229.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
LeDizés, S. Dizés, S. & Fabre, D. 2006 Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices. J. Fluid Mech. (submitted).Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Annu. Rev. Fluid Mech. 10, 221–246.CrossRefGoogle Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769–779.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, 753–763.CrossRefGoogle Scholar
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.CrossRefGoogle Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.CrossRefGoogle Scholar
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251–288.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schmid, P. J., Henningson, D. S., Khorrami, M. R. & Malik, M. R. 1993 A study of eigenvalue sensitivity for hydrodynamic stability operators. Theor. Comp. Fluid Dyn. 4, 227–240.CrossRefGoogle Scholar
Stewartson, K. & Brown, S. N. 1985 Near-neutral centre-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387–399.CrossRefGoogle Scholar
Thomson, W. 1887 (Lord Kelvin) Rectilineal motion of viscous fluid between two parallel planes. Phil. Mag. 24, 188–196.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science. 261, 578–584.CrossRefGoogle ScholarPubMed