Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T11:38:56.025Z Has data issue: false hasContentIssue false
  • English
  • Français

The linear stability of high-frequency flow in a torsionally oscillating cylinder

Published online by Cambridge University Press:  28 March 2007

P. J. BLENNERHASSETT  and
ANDREW P. BASSOM
P. J. BLENNERHASSETT
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia
ANDREW P. BASSOM
Affiliation:
School of Mathematics and Statistics, University of Western Australia, Crawley 6009, Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of the Stokes layer induced in a fluid contained within a long cylinder oscillating at high frequency about its longitudinal axis is investigated. The disturbance equations are derived using Floquet theory and the resulting system solved using pseudo-spectral methods. Both shear modes and axially periodic centripetal disturbance modes are examined and neutral stability curves and corresponding critical conditions for instability identified. For sufficiently small cylinder radius it is verified that the centripetal perturbations limit the stability of the motion but that in larger-radius configurations the shear modes associated with the Stokes layer take over this role. These results suggest a possible design, free of entry-length effects, for experiments intended to examine the breakdown of oscillatory boundary layers.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 576 , 10 April 2007 , pp. 491 - 505
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Akhavan, R., Kamm, R. D. & Shapiro, A. H. 1991 An investigation of transition to turbulence in bounded oscillatory Stokes flows. Part 2. Numerical simulations. J. Fluid Mech. 225, 423–444.CrossRefGoogle Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.CrossRefGoogle Scholar
Benjamin, T. B. & Mullin, T. 1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377, 221–249.Google Scholar
Blennerhassett, P. J. 1976 Secondary motion and diffusion in unsteady flow in a curved pipe. PhD thesis, Department of Mathematics, Imperial College, University of London.Google Scholar
Blennerhassett, P. J. & Bassom, A. P. 2002 The linear stability of flat {S}tokes layers. J. Fluid Mech. 464, 393–410.CrossRefGoogle Scholar
Blennerhassett, P. J. & Bassom, A. P. 2006 The linear stability of high-frequency oscillatory flow in a channel. J. Fluid Mech. 556, 1–25.CrossRefGoogle Scholar
Blennerhassett, P. J. & Hall, P. 1979 Centrifugal instablities of circumferential flows in finite cylinders: linear theory. Proc. R. Soc. Lond. A 365, 191–207.Google Scholar
Blondeaux, P. & Seminara, G. 1979 Transizione incipiente al fondo un'onda di gravità. Rendi. Accad. Naz. Lincei 67, 407–417.Google Scholar
Blondeaux, P. & Vittori, G. 1994 Wall imperfections as a triggering mechanism for {S}tokes-layer transition. J. Fluid Mech. 264, 107–135.CrossRefGoogle Scholar
Cole, J. A. 1976 Taylor-vortex instability and annulus-length effects. J. Fluid Mech. 75, 1–15.CrossRefGoogle Scholar
Costamagna, P., Vittori, G. & Blondeaux, P. 2003 Coherent structures in oscillatory boundary layers. J. Fluid Mech. 474, 1–33.CrossRefGoogle Scholar
Donnelly, R. J. 1964 Experiments on stability of viscous flow between rotating cylinders. III. enhancement of stability by modulation. Proc. R. Soc. Lond. A 281, 130–139.Google Scholar
Fornberg, B. 1996 A Practical Guide to Pseudospectral Methods. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Hall, P. 1978 The linear stability of flat Stokes layers. Proc. R. Soc. Lond. A 359, 151–166.Google Scholar
Kao, T. W. & Park, C. 1970 Experimental investigations of the stability of channel flows. Part 1. Flow of a single liquid in a rectangular channel. J. Fluid Mech. 43, 145–164.CrossRefGoogle Scholar
vonKerczek, C. Kerczek, C. & Davis, S. H. 1974 Linear stability theory of oscillatory {S}tokes layers. J. Fluid Mech. 62, 753–773.Google Scholar
Nishioka, M., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731–751.CrossRefGoogle Scholar
Papageorgiou, D. 1987 Stability of unsteady viscous flow in a curved pipe. J. Fluid Mech. 182, 209–233.CrossRefGoogle Scholar
Reynolds, W. C. & Potter, M. C. 1967 Finite-amplitude instability of parallel shear flows. J. Fluid Mech. 27, 465–492.CrossRefGoogle Scholar
Riley, P. J. & Laurence, R. L. 1976 Linear stability of modulated {C}ouette flow. J. Fluid Mech. 75, 625–646.CrossRefGoogle Scholar
Seminara, G. & Hall, P. 1976 Centrifugal instability of a Stokes layer: linear theory. Proc. R. Soc. Lond. A 350, 299–316.Google Scholar
Smith, F. T. 1975 Pulsatile flow in curved pipes. J. Fluid Mech. 71, 15–42.CrossRefGoogle Scholar
Spalart, P. R. & Baldwin, B. S. 1988 Direct simulation of a turbulent oscillating boundary layer. In Turbulent Shear Flows 6 (ed. André, J., Cousteix, J., Durst, F., Launder, B., Schmidt, F. & Whitelaw, J.). Springer.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
Thomas, L. H. 1953 The stability of plane {P}oiseuille flow. Phys. Rev. 91, 780–783.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. Philadelphia, PA: SIAM.CrossRefGoogle Scholar
Verzicco, R. & Vittori, G. 1996 Direct simulation of transition in {S}tokes boundary layers. Phys. Fluids 8, 1341–1343.CrossRefGoogle Scholar
Vittori, G. & Verzicco, R. 1998 Direct simulation of transition in an oscillatory boundary layer. J. Fluid Mech. 371, 202–232.CrossRefGoogle Scholar
Wu, X. 1992 The nonlinear evolution of high-frequency resonant-triad waves in an oscillatory {S}tokes layer at high {R}eynolds number. J. Fluid Mech. 245, 553–597.CrossRefGoogle Scholar
Yang, W. H. & Yih, C.-S. 1977 Stability of time-periodic flows in a circular pipe. J. Fluid Mech. 82, 497–505.CrossRefGoogle Scholar