Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-28T23:33:01.862Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical calculations of the primary-flow exchange process in the Taylor problem

Published online by Cambridge University Press:  21 April 2006

K. A. Cliffe
K. A. Cliffe
Affiliation:
Theoretical Physics Division, UKAEA Harwell Laboratory, Didcot, Oxon OX11 0RA, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical methods are used to study the way in which the number of cells present in the Taylor experiment changes as the length of the comparatively short annulus varies. The structure of the solution surface is determined by following paths of singular points in a finite-element discretization of the axisymmetric Navier–Stokes equations. The numerical results are compared with the experiments of Benjamin (1978b), Mullin (1982) and Mullin et al. (1982). The calculations are in agreement with the qualitative theory of Benjamin (1978a) and Schaeffer (1980) except that in the interaction involving four- and six-cell flows, the numerical calculations indicate that the six-cell flow can become unstable owing to perturbations that are antisymmetric about the midplane.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 197 , December 1988 , pp. 57 - 79
Copyright
© 1988 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

Benjamin, T. B.1978a Bifurcation phenomena in steady flows of a viscous liquid. I. Theory. Proc. R. Soc. Lond. A 359, 1–26.Google Scholar
Benjamin, T. B.1978b 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.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., Jackson, C. P. & Greenfield, A. C.1982 Finite-element solutions for flow in a symmetric channel with a smooth expansion. Harwell Rep. AERE R-10608. HMSO.Google Scholar
Cliffe, K. A., Jepson, A. & Spence, A.1986 The numerical solution of bifurcation problems with symmetry with application to the finite Taylor problem. In Numerical Methods for Fluid Dynamics II (ed. K. W. Morton & M. J. Baines), pp. 155–176. Clarendon.
Cliffe, K. A. & Mullin, T.1985 A numerical and experimental study of anomalous modes in the Taylor experiment. J. Fluid Mech. 153, 243–258.Google Scholar
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. Mittelmann & H. Weber), pp. 129–144. Birkhäuser: ISNM.
Cliffe, K. A. & Spence, A.1986 Numerical calculations of bifurcations in the finite Taylor problem. In Numerical Methods for Fluid Dynamics II (ed. K. W. Morton & M. J. Baines), pp. 177–197. Clarendon.
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. Num. Meth. Fluids 2, 25–42.Google Scholar
Golubitsky, M. & Schaeffer, D. G.1979 A theory of imperfect bifurcation, via singularity theory. Communs Pure Appl. Maths 32, 21–98.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.1985 Folds in solutions of two parameter systems. SIAM J. Numer. Anal. 22, 347–368.Google Scholar
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.
Kusnetsov, E. A., Lvov, V. S., Nesterikhin, Y. E., Shmojlov, Y. F., Sobolev, V. S., Spector, M. D., Timokhin, S. A., Utkin, E. N. & Vasilenko, Y. G. 1977 About turbulence arising in Couette flows. Inst. Automation and Electrometry, Siberian Branch, USSR Acad. Sci., Preprint no. 58.
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., Pfister, G. & Lorenzen, A.1982 New observations on hysteresis effects in Taylor-Couette flow. Phys. Fluids 25, 1134–1136.Google Scholar
Roberts, P. H.1965 Appendix to Experiments on the stability of viscous flow between rotating cylinders. VI. Finite, amplitude experiments. Proc. R. Soc. Lond. A 283, 531–533.Google Scholar
Schaeffer, D. G.1980 Analysis of a model in the Taylor problem. Math. Proc. Camb. Phil. Soc. 87, 307–337.Google Scholar
Schrauf, G.1986 The first instability in spherical Taylor—Couette flow. J. Fluid Mech. 166, 287–303.Google Scholar
Tuckerman, L. S.1983 Formation of Taylor vortices in spherical Couette flow. Ph.D. thesis, Massachusetts Institute of Technology.
Werner, B.1984 Regular systems for bifurcation points with underlying symmetries. In Numerical Methods for Bifurcation Problems (ed. T. Küpper, H. D. Mittelmann & H. Weber), pp. 562–574. Birkhäuser: ISNM.
Werner, B. & Spence, A.1984 The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399.Google Scholar