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Viscous symmetric stability of circular flows

Published online by Cambridge University Press:  19 May 2010

R. C. KLOOSTERZIEL
R. C. KLOOSTERZIEL*
Affiliation:
School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, HI 96822, USA
*
†Email address for correspondence: [email protected]
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Abstract

The linear stability properties of viscous circular flows in a rotating environment are studied with respect to symmetric perturbations. Through the use of an effective energy or Lyapunov functional, we derive sufficient conditions for Lyapunov stability with respect to such perturbations. For circular flows with swirl velocity V(r) we find that Lyapunov stability is determined by the properties of the function ℱ(r) = (2V/r + f)/Q (with f the Coriolis parameter, r the radius and Q the absolute vorticity) instead of the customary Rayleigh discriminant Φ(r) = (2V/r + f)Q. The conditions for stability are valid for flows with non-zero Q everywhere. Further, the flows are presumed stationary, incompressible and velocity perturbations are required to vanish at rigid boundaries. For Lyapunov stable flows an upper bound for the increase of the total perturbation energy due to transient non-modal growth is derived which is valid for any Reynolds number. The theory is applied to Couette flow and the Lamb–Oseen vortex.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 171 - 193
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Arnol'd, V. I. 1966 On an a priori estimate in the theory of hydrodynamical stability. Izv. Vyssh. Uchebn. Zaved. Mat. 54, (5), 3–5. Translation in 1969 Am. Math. Soc. Transl. Ser. 2 79, 267–269.Google Scholar
Bayly, B. J. 1988 Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31, 56–64.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.CrossRefGoogle Scholar
Carnevale, G. F., Briscolini, M., Kloosterziel, R. C. & Vallis, G. K. 1997 Three-dimensionally perturbed vortex tubes in a rotating flow. J. Fluid Mech. 341, 127–163.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Charney, J. G. 1973 Lecture notes on planetary fluid dynamics. In Dynamic Meteorology (ed. Morel, P.), pp. 97–352. Reidel.CrossRefGoogle Scholar
Cho, H.-R., Shepherd, T. G. & Vladimirov, V. A. 1993 Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. J. Atmos. Sci. 50, 822–836.2.0.CO;2>CrossRefGoogle Scholar
Drazin, P. & Reid, W. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Eliassen, A. 1951 Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norvegia 5, 19–60.Google Scholar
Eliassen, A. & Kleinschmidt, E. 1957 Dynamic meteorology. In Handbuch der Physik (ed. Bartels, J.), vol. 48, pp. 1–154. Springer.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, (8), 2093–2102.CrossRefGoogle Scholar
Fjørtoft, R. 1950 Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic vortex. Geofys. Publ. 17, (6), 5–52.Google Scholar
Høiland, E. 1962 Discussion of a hyperbolic equation relating to inertia and gravitational fluid oscillations. Geofys. Publ. 24, 211–227.Google Scholar
Kloosterziel, R. C. & Carnevale, G. F. 2007 Generalized energetics for inertially stable parallel shear flows. J. Fluid Mech. 585, 117–126.CrossRefGoogle Scholar
Kloosterziel, R. C., Carnevale, G. F. & Orlandi, P. 2007 Inertial instability in rotating and stratified fluids: barotropic vortices. J. Fluid Mech. 583, 379–412.CrossRefGoogle Scholar
Kloosterziel, R. C. & van Heijst, G. J. F. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.CrossRefGoogle Scholar
McIntyre, M. E. 1970 Diffusive destabilization of the baroclinic circular vortex. Geophys. Fluid Dyn. 1, 19–58.CrossRefGoogle Scholar
van Mieghem, J. M. 1951 Hydrodynamic instability. In Compendium of Meteorology (ed. Malone, T. F.), pp. 434–453. American Meteorological Society.CrossRefGoogle Scholar
Miyazaki, T. & Hunt, J. C. R. 2000 Linear and nonlinear interactions between a columnar vortex and external turbulence. J. Fluid Mech. 402, 349–378.CrossRefGoogle Scholar
Mu, M., Shepherd, T. G. & Swanson, K. 1996 On nonlinear symmetric stability and the nonlinear saturation of symmetric instability. J. Atmos. Sci 53, 2918–2923.2.0.CO;2>CrossRefGoogle Scholar
Ooyama, K. 1966 On the stability of the baroclinic circular vortex: a sufficient condition for instability. J. Atmos. Sci. 23, 43–53.2.0.CO;2>CrossRefGoogle Scholar
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid. Mech. 550, 251–288.CrossRefGoogle Scholar
Rayleigh, J. W. S. 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Sawyer, S. J. 1947 Notes on the theory of tropical cyclones. Quart. J. R. Met. Soc. 73, 101–126.CrossRefGoogle Scholar
Sawyer, S. J. 1949 The significance of dynamic instability in atmospheric motions. Quart. J. R. Met. Soc. 75, 364–374.CrossRefGoogle Scholar
Schmidt, P. J. 2007 Nonmodal stability theory. Ann. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 2000 Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows in rotating systems. Phys. Fluids A 12, 1740–1748.CrossRefGoogle Scholar
Solberg, H. 1936 Le mouvement d'inertie de l'atmosphere stable et son role dans la theorie des cyclones. Sixth Assembly, Edinburgh. Union Geodesique et Geophysique Internationale, 66–82.Google Scholar
Synge, J. L. 1938 On the stability of a viscous liquid between rotating coaxial cylinders. Proc. Roy. Soc. A 167, 250–256.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.CrossRefGoogle ScholarPubMed
Wood, W. W. 1964 Stability of viscous flow between rotating cylinders. ZAMP 15, (3), 313–314.Google Scholar
Yanai, M. & Tokiaka, T. 1969 Axially symmetric meridional motions in the baroclinic circular vortex: a numerical experiment. J. Meteorol. Soc. Japan 47, (3), 183–197.CrossRefGoogle Scholar