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Suction–shear–Coriolis instability in a flow between parallel plates

Published online by Cambridge University Press:  04 November 2014

Kengo Deguchi ,
Naoyoshi Matsubara  and
Masato Nagata
Kengo Deguchi*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto, 606-8501, Japan
Naoyoshi Matsubara
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto, 606-8501, Japan
Masato Nagata
Affiliation:
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Kyoto, 606-8501, Japan Department of Mechanics, Tianjin University, Tianjin 300072, PR China
*
† Email addresses for correspondence: [email protected], [email protected]
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Abstract

A rotating fluid flow between differentially translating parallel plates, which induce uniform suction and injection, is studied as a canonical model of swirling flow where suction, shear and Coriolis effects compete. This relatively simple modelling yields several reduced equations that are valid for asymptotically large suction, shear and/or rotation rates. The linear stability problems derived from the full Navier–Stokes and reduced problems are numerically solved and compared. In addition to Taylor-vortex modes, transverse-roll-type instabilities are found in Rayleigh-stable and -unstable parameter regions when weak suction is applied. These instabilities, separated by the so-called Rayleigh line, are characterised by vortices attached to the suction wall. Another type of instability, which exists beyond the Rayleigh line and shows inviscid motion in the fluid core, is found when suction is sufficiently strong. The relation of this instability to the stability results by Gallet, Doering & Spiegel (Phys. Fluids, vol. 22, 2010, 034105) is discussed. Our nonlinear analyses indicate subcritical and supercritical bifurcations of finite-amplitude solutions for the near-wall and fluid-core instabilities, respectively.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 760 , 10 December 2014 , pp. 212 - 242
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Copyright
© 2014 Cambridge University Press 

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References

Avila, M. 2012 Stability and angular-momentum transport of fluid flows between corotating cylinders. Phys. Rev. Lett. 108, 124501.CrossRefGoogle ScholarPubMed
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Cowley, S. J. & Smith, F. T. 1985 On the stability of Poiseuille–Couette flow: a bifurcation from infinity. J. Fluid Mech. 156, 83–100.CrossRefGoogle Scholar
Deguchi, K. & Hall, P. 2014 Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752, 602–625.CrossRefGoogle Scholar
Deguchi, K., Hall, P. & Walton, A. G. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 721, 58–85.Google Scholar
Deguchi, K., Meseguer, A. & Mellibovsky, F. 2014 Subcritical equilibria in Taylor–Couette flow. Phys. Rev. Lett. 112, 184502.Google Scholar
Deguchi, K. & Walton, A. G. 2013 Axisymmetric travelling waves in annular sliding Couette flow at finite and asymptotically large Reynolds number. J. Fluid Mech. 720, 582–617.CrossRefGoogle Scholar
Doering, C. R., Spiegel, E. A. & Worthing, R. A. 2000 Energy dissipation in a shear layer with suction. Phys. Fluids 12 (8), 1955–1968.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dubrulle, B., Dauchot, O., Daviaud, F., Longaretti, P.-Y., Richard, D. & Zahn, J.-P. 2005 Stability and turbulent transport in Taylor–Couette flow from analysis of experimental data. Phys. Fluids 17 (9), 095103.Google Scholar
Fransson, J. H. M. & Alfredsson, P. H. 2003a On the disturbance growth in an asymptotic suction boundary layer. J. Fluid Mech. 482, 51–90.Google Scholar
Fransson, J. H. M. & Alfredsson, P. H. 2003b On the hydrodynamic stability of channel flow with cross-flow. Phys. Fluids 15 (2), 436–441.Google Scholar
Gallet, B., Doering, C. R. & Spiegel, E. A. 2010 Destabilizing Taylor–Couette flow with suction. Phys. Fluids 22, 034105.Google Scholar
Guha, A. & Frigaard, I. A. 2010 On the stability of plane Couette–Poiseuille flow with uniform crossflow. J. Fluid Mech. 656, 417–447.Google Scholar
Hains, F. D. 1971 Stability of plane Couette–Poiseuille flow with uniform cross-flow. Phys. Fluids 14, 1620–1623.CrossRefGoogle Scholar
Hall, P. 1988 The nonlinear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 193, 243–266.Google Scholar
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178–205.CrossRefGoogle Scholar
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641–666.CrossRefGoogle Scholar
Hocking, L. M. 1975 Nonlinear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths 28 (3), 341–353.Google Scholar
Ji, H., Burin, M., Schartman, E. & Goodman, J. 2006 Hydrodynamic turbulence cannot transport angular momentum effectively in astrophysical disks. Nature 444, 343–346.Google Scholar
Johnston, J. P., Halleen, R. M. & Lezius, D. K. 1972 Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J. Fluid Mech. 56, 533–557.Google Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2013 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, R6.Google Scholar
Lahey, P. M. & Drew, D. A. 2002 Taylor vortex flow with suction. Chem. Engng Sci. 57 (24), 5161–5173.Google Scholar
Luchini, P. 1996 Reynolds-number-independent instability of the boundary layer over a flat surface. J. Fluid Mech. 327, 101–115.Google Scholar
Messing, R. & Kloker, M. J. 2010 Investigation of suction for laminar flow control of three-dimensional boundary layers. J. Fluid Mech. 658, 117–147.Google Scholar
Milinazzo, F. A. & Saffman, P. G. 1985 Finite-amplitude steady waves in plane viscous shear flows. J. Fluid Mech. 160, 281–295.CrossRefGoogle Scholar
Min, K. & Lueptow, R. M. 1994 Hydrodynamic stability of viscous flow between rotating porous cylinders with radial flow. Phys. Fluids 6 (1), 144–151.Google Scholar
Nagata, M. 1986 Bifurcations in Couette flow between almost corotating cylinders. J. Fluid Mech. 169, 229–250.Google Scholar
Nicoud, F. & Angilella, J. R. 1997 Effects of uniform injection at the wall on the stability of Couette-like flows. Phys. Rev. E 56 (3), 3000–3009.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50 (4), 689–703.Google Scholar
Paoletti, M. S. & Lathrop, D. P. 2011 Angular momentum transport in turbulent flow between independently rotating cylinders. Phys. Rev. Lett. 106, 024501.Google Scholar
Park, D. S. & Huerre, P. 1995 Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plane. J. Fluid Mech. 283, 249–272.Google Scholar
Rayleigh, Lord 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148–154.Google Scholar
Rincon, F., Ogilvie, G. I. & Cossu, C. 2007 On self-sustaining processes in Rayleigh-stable rotating plane Couette flows and subcritical transition to turbulence in accretion disks. Astron. Astrophys. 463, 817–832.Google Scholar
Sheppard, D. M. 1972 Hydrodynamic stability of the flow between parallel porous walls. Phys. Fluids 15, 241–244.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Proc. R. Soc. Lond. A 223, 289–343.Google Scholar