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Stochastic forcing of the Lamb–Oseen vortex

Published online by Cambridge University Press:  01 October 2008

J. FONTANE
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université Paul Sabatier, 2 Allée du Professeur Camille Soula, 31400 Toulouse, France
P. BRANCHER
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université Paul Sabatier, 2 Allée du Professeur Camille Soula, 31400 Toulouse, France
D. FABRE
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université Paul Sabatier, 2 Allée du Professeur Camille Soula, 31400 Toulouse, France

Abstract

The aim of the present paper is to analyse the dynamics of the Lamb–Oseen vortex when continuously forced by a random excitation. Stochastic forcing is classically used to mimic external perturbations in realistic configurations, such as variations of atmospheric conditions, weak compressibility effects, wing-generated turbulence injected into aircraft wakes, or free-stream turbulence in wind tunnel experiments. The linear response of the Lamb–Oseen vortex to stochastic forcing can be decomposed in relation to the azimuthal symmetry of the perturbation given by the azimuthal wavenumber m. In the axisymmetric case m = 0, we find that the response is characterized by the generation of vortex rings at the outer periphery of the vortex core. This result is consistent with recurrent observations of such dynamics in the study of vortex–turbulence interaction. When considering helical perturbations m = 1, the response at large axial wavelengths consists of a global translation of the vortex, a feature very similar to the phenomenon of vortex meandering (or wandering) observed experimentally, corresponding to an erratic displacement of the vortex core. At smaller wavelengths, we find that stochastic forcing can excite specific oscillating modes of the Lamb–Oseen vortex. More precisely, damped critical-layer modes can emerge via a resonance mechanism. For perturbations with higher azimuthal wavenumber m ≥ 2, we find no structure that clearly dominates the response of the vortex.

Type
Papers
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Antkowiak, A. 2005 Dynamique aux temps courts d'un tourbillon isolé. PhD thesis, Université Paul Sabatier (UPS), Toulouse, France.Google Scholar
Antkowiak, A. & Brancher, P. 2004 Transient energy growth for the Lamb-Oseen vortex. Phys. Fluids 16, L1L4.CrossRefGoogle Scholar
Antkowiak, A. & Brancher, P. 2007 On vortex rings around vortices: an optimal mechanism. J. Fluid Mech. 578, 295304.CrossRefGoogle Scholar
Baker, G. R., Barker, S. J., Bofah, K. K. & Saffman, P. G. 1974 Laser anemometer measurements of trailing vortices in water. J. Fluid Mech. 65, 325336.CrossRefGoogle Scholar
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13, 32583269.CrossRefGoogle Scholar
Billant, P., Brancher, P. & Chomaz, J. M. 1999 Three-dimensional stability of a vortex pair. Phys. Fluids 11, 20692077.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 16371650.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5, 774777.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2001 Optimal linear growth in swept bounday layers. J. Fluid Mech. 435, 123.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 21722179.CrossRefGoogle Scholar
Devenport, W. J., Rife, M. C., Liapis, S. I. & Follin, G. J. 1996 The structure and development of a wing-tip vortex. J. Fluid Mech. 312, 60106.CrossRefGoogle 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
Farrell, B. F. & Ioannou, P. J. 1993 a Optimal excitation of three-dimensionnal perturbations in viscous constant shear flow. Phys. Fluids A 5, 13901400.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 b Stochastic forcing of the linearized Navier-Stokes equations. Phys. Fluids A 5, 26002609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1994 Variance maintained by stochasting forcing of non-normal dynamical systems associated with linearly stable shear flows. Phys. Rev. Lett. 72, 11881191.CrossRefGoogle ScholarPubMed
Hoepffner, J. 2006 Stability and control of shear flows subject to stochastic excitations. PhD thesis, R. Inst. Tech. (KTH), Stockholm, Sweden.Google Scholar
Joly, L., Fontane, J. & Chassaing, P. 2005 The Rayleigh-Taylor instability of two-dimensional high-density vortices. J. Fluid Mech. 537, 415431.CrossRefGoogle Scholar
Jovanovic, M. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145183.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85119.CrossRefGoogle Scholar
Marshall, J. S. & Beninati, M. L. 2005 External turbulence interaction with a columnar vortex. J. Fluid Mech. 540, 221245.CrossRefGoogle Scholar
Melander, M. V. & Hussain, F. 1993 Coupling between a coherent structure and fine-scale turbulence. Phys. Rev. E 48, 26692689.Google ScholarPubMed
Miyazaki, T. & Hunt, J. 2000 Linear and non-linear interactions between a columnar vortex and external turbulence. J. Fluid Mech. 402, 349378.CrossRefGoogle Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straigth vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413425.Google Scholar
Nolan, D.S. & Farrell, B. F. 1999 The intensification of two-dimensional swirling flows by stochastic asymmetric forcing. J. Atmos. Sci. 56, 39373962.2.0.CO;2>CrossRefGoogle Scholar
Orr, W. M. 1907 a The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. part 1: A perfect liquid. Proc. R. Irish Acad. 27, 968.Google Scholar
Orr, W. M. 1907 b The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. part 2: A viscous liquid. Proc. R. Irish Acad. 27, 69138.Google Scholar
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251288.CrossRefGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr-Sommerfeld operator. SIAM J. Appl. Maths 53, 1547.CrossRefGoogle Scholar
Risso, F., Corjon, A. & Stoessel, A. 1997 Direct numerical simulations of wake vortices in intense homogeneous turbulence. AIAA J. 35, 10301040.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.CrossRefGoogle Scholar
Takahashi, N., Ishii, H. & Miyazaki, T. 2005 The influence of turbulence on a columnar vortex. Phys. Fluids 17, 035105.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578584.CrossRefGoogle ScholarPubMed
Tsai, C. Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed starin field. J. Fluid Mech. 73, 721733.CrossRefGoogle Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A matlab differentiation matrix suite. ACM Trans. Math. Soft. 26, 465519.CrossRefGoogle Scholar
Whitaker, J. S. & Sardeshmukh, P. D. 1998 A linear theory of extratropical synoptic eddy statistics. J. Atmos. Sci. 55, 238258.2.0.CO;2>CrossRefGoogle Scholar
Zhou, K., Doyle, J. & Glover, K. 1995 Robust and Optimal Control, 1st edn. Prentice Hall.Google Scholar