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Instabilities and transient growth of the stratified Taylor–Couette flow in a Rayleigh-unstable regime

Published online by Cambridge University Press:  31 May 2017

Junho Park Open the ORCID record for Junho Park [Opens in a new window] ,
Paul Billant  and
Jong-Jin Baik
Junho Park*
Affiliation:
School of Earth and Environmental Sciences, Seoul National University, Seoul 151-742, Republic of Korea
Paul Billant
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, F-91128 Palaiseau CEDEX, France
Jong-Jin Baik
Affiliation:
School of Earth and Environmental Sciences, Seoul National University, Seoul 151-742, Republic of Korea
*
†Email address for correspondence: [email protected]
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Abstract

The stability of the Taylor–Couette flow is analysed when there is a stable density stratification along the axial direction and when the flow is centrifugally unstable, i.e. in the Rayleigh-unstable regime. It is shown that not only the centrifugal instability but also the strato-rotational instability can occur. These two instabilities can be explained and well described by means of a Wentzel–Kramers–Brillouin–Jeffreys asymptotic analysis for large axial wavenumbers in inviscid and non-diffusive limits. In the presence of viscosity and diffusion, numerical results reveal that the strato-rotational instability becomes dominant over the centrifugal instability at the onset of instability when the axial density stratification is sufficiently strong. Linear transient energy growth is next investigated for counter-rotating cylinders in the stable regime of the Froude number–Reynolds number parameter space. We show that there exist two types of transient growth mechanism analogous to the lift up and the Orr mechanisms in homogeneous fluids but with the additional effect of density perturbations. The dominant mechanism depends on the stratification: when the stratification is strong, non-axisymmetric three-dimensional perturbations achieve the optimal energy growth through the Orr mechanism while for moderate stratification, axisymmetric perturbations lead to the optimal transient growth by a lift-up mechanism involving internal waves.

JFM classification

Instability: Instability Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 80 - 108
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© 2017 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Agbessi, Y., Alibenyahia, B., Nouar, C., Hof, L. C. B. & Avila, M. 2015 Linear stability of Taylor–Couette flow of shear-thinning fluids: modal and non-modal approaches. J. Fluid Mech. 776, 354–389.Google Scholar
Andereck, C. D., Liu, S. S. & Swinney, H. L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.Google Scholar
Antkowiak, A.2005 Dynamique aux temps courts d’un tourbillon isolé. PhD thesis, Université Paul Sabatier de Toulouse.Google Scholar
Arratia, C.2011 Non-modal instability mechanisms in stratified and homogeneous shear flow. PhD thesis, Ecole Polytechnique.Google Scholar
Bakas, N. A. & Farrell, B. F. 2009a Gravity waves in a horizontal shear flow. Part I: growth mechanisms in the absence of potential vorticity perturbations. J. Phys. Oceanogr. 39, 481–496.Google Scholar
Bakas, N. A. & Farrell, B. F. 2009b Gravity waves in a horizontal shear flow. Part II: interaction between gravity waves and potential vorticity perturbations. J. Phys. Oceanogr. 39, 497–511.CrossRefGoogle Scholar
Bakas, N. A., Ioannou, P. J. & Kefaliakos, G. E. 2001 The emergence of coherent structures in stratified shear flow. J. Atmos. Sci. 58, 2790–2806.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.Google Scholar
Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.Google Scholar
Billant, P. & Gallaire, F. 2013 A unified criterion for the centrifugal instabilities of vortices and swirling jets. J. Fluid Mech. 734, 5–35.Google Scholar
Billant, P. & Le Dizès, S. 2009 Waves on a columnar vortex in a strongly stratified fluid. Phys. Fluids 21, 106602.CrossRefGoogle Scholar
Boubnov, B. M., Gledzer, E. B. & Hopfinger, E. J. 1995 Stratified circular Couette flow: instability and flow regimes. J. Fluid Mech. 292, 333–358.Google Scholar
Caton, F., Janiaud, B. & Hopfinger, E. J. 2000 Stability and bifurcations in stratified Taylor–Couette flow. J. Fluid Mech. 419, 93–124.Google Scholar
Chen, C., Wan, Z.-H. & Zhang, W.-G. 2015 Transient growth in Taylor–Couette flow of a Bingham fluid. Phys. Rev. E 91, 043202.Google ScholarPubMed
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Dubrulle, B., Dauchot, O., Daviaud, F., Longaretti, P.-Y., Richard, D. & Zahn, J.-P. 2005a Stability and turbulent transport in Taylor–Couette flow from analysis of experimental data. Phys. Fluids 17, 095103.CrossRefGoogle Scholar
Dubrulle, B., Marié, L., Normand, C., Richard, D., Hersant, F. & Zahn, J.-P. 2005b An hydrodynamic stability shear instability in stratified disks. Astron. Astrophys. 29, 1–13.Google Scholar
Eaves, T. S. & Caulfield, C. P. 2015 Disruption of SSP/VWI states by a stable stratification. J. Fluid Mech. 784, 548–564.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Transient development of perturbations in stratified shear flow. J. Atmos. Sci. 50, 2201–2214.Google Scholar
Grossmann, S., Lohse, D. & Sun, C. 2016 High-Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech. 48, 53–80.Google Scholar
Hitchman, H. M., Leovy, C. B., Gille, J. C. & Bailey, P. K. 1987 Quasi-stationary zonally asymmetric circulations in the equatorial lower mesosphere. J. Atmos. Sci. 44, 2219–2236.Google Scholar
Hristova, H., Roch, S., Schmid, P. J. & Tuckerman, S. L. 2002 Transient growth in Taylor–Couette flow. Phys. Fluids 14, 3474–3484.Google Scholar
Hua, B. L., Le Gentil, S. & Orlandi, P. 1997a First transitions in circular Couette flow with axial stratification. Phys. Fluids 9, 365–375.Google Scholar
Hua, B. L., Moore, D. W. & Le Gentil, S. 1997b Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech. 331, 345–371.Google Scholar
Kaminski, A. K., Caulfield, C. P. & Taylor, J. R. 2014 Transient growth in strongly stratified shear layers. J. Fluid Mech. 758, R4.CrossRefGoogle Scholar
Le Bars, M. & Le Gal, P. 2007 Experimental analysis of the stratorotational instability in a cylindrical Couette flow. Phys. Rev. Lett. 99, 064502.Google Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.Google Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.CrossRefGoogle Scholar
Le Dizès, S. & Riedinger, X. 2010 The strato-rotational instability of Taylor–Couette and Keplerian flows. J. Fluid Mech. 660, 147–161.Google Scholar
Leclerq, C., Nguyen, F. & Kerswell, R. R. 2016 Connections between centrifugal, stratorotational, and radiative instabilities in viscous Taylor–Couette flow. Phys. Rev. E 94, 043103.Google Scholar
Maretzke, S., Hof, B. & Avila, M. 2014 Transient growth in linearly stable Taylor–Couette flows. J. Fluid Mech. 742, 254–290.Google Scholar
Meseguer, A. 2002 Energy transient growth in the Taylor–Couette problem. Phys. Fluids 14, 1655–1660.Google Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2001 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270–5273.Google Scholar
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106, 134502.Google Scholar
Olver, F. W. J. 1974 Asymptotics and Special Functions. Academic.Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: a viscous liquid. Proc. R. Irish Acad. A 27, 69–138.Google Scholar
Park, J.2012 Waves and instabilities on vortices in stratified and rotating fluids. PhD thesis, Ecole Polytechnique.Google Scholar
Park, J. & Billant, P. 2012 Radiative instability of an anticyclonic vortex in a stratified rotating fluid. J. Fluid Mech. 707, 381–392.Google Scholar
Park, J. & Billant, P. 2013a Instabilities and waves on a columnar vortex in a strongly stratified and rotating fluid. Phys. Fluids 25, 086601.Google Scholar
Park, J. & Billant, P. 2013b The stably stratified Taylor–Couette flow is always unstable except for solid-body rotation. J. Fluid Mech. 725, 262–280.Google Scholar
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244–272.Google Scholar
Rayleigh, Lord 1917 On the dynamics of revolving fluids. Phil. Trans. R. Soc. Lond. A 93, 148–154.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Shalybkov, D. & Rüdiger, G. 2005 Stability of density-stratified viscous Taylor–Couette flows. Astron. Astrophys. 438, 411–417.CrossRefGoogle Scholar
Smyth, W. D. & McWilliams, J. C. 1998 Instability of an axisymmetric vortex in a stably stratified, rotating environment. Theor. Comput. Fluid Dyn. 11, 305–322.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Withjack, E. M. & Chen, C. F. 1974 An experimental study of Couette instability of stratified fluids. J. Fluid Mech. 66, 725–737.Google Scholar
Withjack, E. M. & Chen, C. F. 1975 Stability analysis of rotational Couette flow of stratified fluids. J. Fluid Mech. 68, 157–175.Google Scholar
Yavneh, I., McWilliams, J. C. & Molemaker, M. J. 2001 Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow. J. Fluid Mech. 448, 1–21.Google Scholar
Yim, E. & Billant, P. 2016 Analogies and differences between the stability of an isolated pancake vortex and a columnar vortex in stratified fluid. J. Fluid Mech. 796, 732–766.Google Scholar