Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-02-05T07:11:02.951Z Has data issue: false hasContentIssue false
  • English
  • Français

A numerical and experimental study of anomalous modes in the Taylor experiment

Published online by Cambridge University Press:  20 April 2006

K. A. Cliffe  and
T. Mullin
K. A. Cliffe
Affiliation:
Theoretical Physics Division, AERE Harwell, Didcot. Oxon OX 11 ORA
T. Mullin
Affiliation:
Clarendon Laboratory, Parks Road, Oxford OX1 3PU
Get access Rights & Permissions [Opens in a new window]

Abstract

Anomalous modes are flows in the Taylor experiment that exist only for sufficiently high Reynolds number R and are always distinct from the primary flow produced by gradually increasing R from small values. They are distinguished from all other secondary modes by having a direction of spiralling of one or both of the end cells such that outward flow is found along the stationary endwall. In this paper we present new observations of these flows and compare them with numerical solutions of the Navier–Stokes equations. A numerical technique for calculating anomalous modes is described and stability curves for 2-, 3-, and 4-cell flows are presented. Streamline plots of the numerical solutions are compared with photographs of the observed flows. The agreement between the calculations and experiments is good. The calculations also confirm certain theoretical predictions made by Benjamin (1978), Benjamin & Mullin (1981) and Hall (1982).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 153 , April 1985 , pp. 243 - 258
Copyright
© 1985 Cambridge University Press

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

Bauer, L., Keller, H. B. & Reiss, E. L. 1975 Multiple eigenvalues lead to secondary bifurcation. SIAM Rev. 17 101–122.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous liquid. II. Experiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Benjamin, T. B. & Mullin, T. 1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377, 221–249.Google Scholar
Benjamin, T. B. & Mullin, T. 1982 Notes on the multiplicity of flows in the Taylor experiment. J. Fluid Mech. 121, 219–230.Google Scholar
Cliffe, K. A. 1983 Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233.Google Scholar
Cliffe, K. A. 1984 Numerical calculations of the primary flow exchange process in the Taylor problem. Submitted to J. Fluid Mech.Google Scholar
Cliffe, K. A., Jackson, C. P. & Greenfield, A. C. 1982 Finite-element solutions for flow in a symmetric channel with a smooth expansion. Harwell report AERE R-10608. HMSO.
Cliffe, K. A. & Spence, A. 1984 The calculation of high order singularities in the finite Taylor problem. In Numerical Methods for Bifurcation Problems (ed. T. Küpper, H. D. Mittlemann & H. Weber). Birkhauser: ISNM.
Cliffe, K. A. & Winters, K. H. 1984 A numerical study of the cusp catastrophe for Bénard convection in tilted cavities. J. Comp. Phys. 54, 531–534.Google Scholar
Engleman, M. S., Sani, R. L., Gresho, P. M. & Bercovier, M. 1982 Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements. Intl J. Numer. Methods Fluids 2, 25–42.Google Scholar
Hall, P. 1980 Centrifugal instabilities in finite containers: a periodic model. J. Fluid Mech. 99, 575–596.Google Scholar
Hall, P. 1982 Centrifugal instabilities of circumferential flows in finite cylinders: the wide gap problem. Proc. R. Soc. Lond. A 384, 359–379.Google Scholar
Jepson, A. & Spence, A. 1984 Folds in solutions of two parameter systems. SIAM J. Numer. Anal. (To appear.)
Keller, H. B. 1977 Numerical solutions of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. P. H. Rabinowitz), pp. 359–384. Academic.
Lorenzen, A. & Mullin, T. 1984 Anomalous modes and finite length effects in Taylor-Couette flow. Submitted to Phys. Rev. A.
Moore, G. & Spence, A. 1980 The calculation of turning points of nonlinear equations. SIAM J. Numer. Anal. 17, 567–576.Google Scholar
Mullin, T. 1982 Mutations of steady cellular flows in the Taylor experiment. J. Fluid Mech. 121, 207–218.Google Scholar
Mullin, T., Cliffe, K. A. & Benjamin, T. B. 1984 In preparation.
Mullin, T. & Lorenzen, A. 1984 Bifurcation phenomena between a rotating circular cylinder and a stationary square outer cylinder. Submitted to J. Fluid Mech.Google Scholar
Schaeffer, D. G. 1980 Analysis of a model in the Taylor problem. Math. Proc. Camb. Phil. Soc. 87, 307–337.Google Scholar
Shearer, M. 1980 Secondary bifurcation near a double eigenvalue. SIAM J. Math. Anal. 11, 365–389.Google Scholar
Werner, B. & Spence, A. 1984 The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399.Google Scholar