Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-18T10:10:11.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Streamline topology in eccentric Taylor vortex flow

Published online by Cambridge University Press:  26 April 2006

P. Ashwin  and
G. P. King
P. Ashwin
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
G. P. King
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Department of Engineering, University of Warwick, Coventry CV4 7AL, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate an asymptotic model of DiPrima & Stuart (1972b, 1975) describing steady Taylor vortex flow between eccentric cylinders, under the assumption that the eccentricity ε, the clearance ratio δ and the Taylor vortex amplitude A satisfy ε, δ and A small. By solving a boundary value problem for the radial eigenfunctions we numerically obtain the flow field of DiPrima & Stuart and investigate its topology, after correcting higher-order terms to ensure that the flow preserves volume. We find regions of chaotic streamlines at all eccentricities and discuss the reason for their existence. We make an analogy between the full model and a modulated vortex flow field which qualitatively displays the same behaviour.

For large eccentricities, we examine the flow field and the topology of its streamlines, especially where the two-dimensional flow contains a separated region of recirculation. In this case Taylor vortices give rise to transport of fluid particles in and out of the separated region. We find that the onset of Taylor vortices encourages recirculation in the inflow plane, whilst discouraging it in the outflow plane.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 285 , 25 February 1995 , pp. 215 - 247
Copyright
© 1995 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.)

Footnotes

With Appendix B by G. Rowlands.

References

Acharya, N., Sen, M. & Chang, H.-C. 1992 Intl J. Heat Mass Transfer 35, 2475–2489.Google Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.Google Scholar
Aref, H. 1990 Chaotic advection of fluid particles. Phil. Trans. R. Soc. Lond. A 333, 273–288.Google Scholar
Arnol'd, V. 1965 Sur la topologie des ecoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris 261, 17–20.Google Scholar
Arrowsmith, D. K. & Place, C. M. 1990 Introduction to Dynamical Systems. Cambridge University Press.Google Scholar
Arter, W. 1983 Ergodic stream-lines in steady convection. Phys. Lett A 97, 171–174.Google Scholar
Ballal, B. Y. & Rivlin, R. S. 1977 Flow of a Newtonian fluid between eccentric rotating cylinders: Inertial effects. Arch. Rat. Mech. Anal. 62, 237–294.Google Scholar
Chirikov, B. V. 1979 A universal unstability of many-dimensional oscillator systems. Phys. Rep. 52, 265–379.Google Scholar
Cole, J. A. 1965 Experiments on Taylor vortices between eccentric rotating cylinders. Proc. 2nd Aust. Conf. Hydr. Fluid Mech.Google Scholar
Cole, J. A. 1976 Taylor-vortex instability and annulus length effects. J. Fluid Mech. 75, 1–15.Google Scholar
Dai, R.-X., Dong, Q., Szeri, A. Z. 1992 Flow between eccentric rotating cylinders: bifurcation and stability. Intl J. Engng Sci. 30, 1323–1340.Google Scholar
DiPrima, R. C. 1963 A note on the stability of flow in loading journal bearings. Am. Soc. Lub. Engrs Trans. 6, 249–253.Google Scholar
DiPrima, R. C. & Stuart, J. T. 1972a Flow between eccentric rotating cylinders. J. Lub. Tech. Trans. ASME: F94, 266–274.Google Scholar
DiPrima, R. C. & Stuart, J. T. 1972b Non-local effects in the stability of flow between eccentric rotating cylinders. J. Fluid Mech. 54, 393–415.Google Scholar
DiPrima, R. C. & Stuart, J. T. 1975 The nonlinear calculation of Taylor-vortex flow between eccentric rotating cylinders. J. Fluid Mech. 67, 85–111.Google Scholar
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353–391.Google Scholar
Eagles, P. M., Stuart, J. T. & DiPrima, R. C. 1975 The effects of eccentricity on torque and load in Taylor-vortex flow. J. Fluid Mech. 87, 209–231.Google Scholar
Hénon, M. 1965 Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris 262, 312–314.Google Scholar
Jones, S. W., Thomas, O. M. & Aref, H. 1989 Chaotic advection by laminar flow in a twisted pipe. J. Fluid Mech. 209, 335–357.Google Scholar
Koschmieder, E. L. 1976 Taylor vortices between eccentric cylinders. Phys. Fluids 19, 1–4.Google Scholar
Karasudani, T. 1987 Non-axis-symmetric Taylor vortex flow in eccentric rotating cylinders. J. Phys. Soc. Japan 56, 855.Google Scholar
Lichtenberg, A. J. & Liebermann, M. A. 1982 Regular and Stochastic Motion. Springer.CrossRefGoogle Scholar
MacKay, R. S. 1994 Transport in 3d volume-preserving flows. J. Nonlinear Sci. 4, 329–354.Google Scholar
Marcus, P. S. 1981 Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241–255.Google Scholar
O'Brien, K. T., Jones, C. D. & Mobbs, F. R. 1974 Separation and cavitation in superlaminar flow between eccentric rotating cylinders. In Leeds–Lyon symposium on Tribology, pp. 69–72.Google Scholar
Oikawa, M., Karasudani, T. & Funakoshi, M. 1989a Stability of flow between eccentric cylinders with a wide gap. J. Phys. Soc. Japan 58, 2209–2210.Google Scholar
Oikawa, M., Karasudani, T. & Funakoshi, M. 1989b Stability of flow between eccentric rotating cylinders. J. Phys. Soc. Japan 58, 2355–2364.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing; Stretching, Chaos and Transport. Cambridge University Press.Google Scholar
Ozogan, M. S. & Mobbs, F. R. 1980 Superlaminar flow between eccentric rotating cylinders at small clearance ratios. In Energy Conservation Through Fluid Film Lubrication Technology: Frontiers in Research and Design (ed. S. M. Rohde, D. F. Wilcock & H. S. Cheng). The American Society of Engineers.Google Scholar
Prandtl, L. 1905 Verhandlungen des III Internationalen Mathematiker–Kongresses (Heidelberg 1904), Leipzig, pp. 484–491. (Also in Prandtl, L. 1961, Gesammelte Abhandlungen, vol. 2, pp. 575–584. Springer.)Google Scholar
Raffaï, R. & Laure, P. 1991 Effets de l'excentrement des cylindres sur les premières bifurcations du problème de Couette–Taylor. C. R. Acad. Sci. Paris 313, 179–184.Google Scholar
Rowlands, G. 1990 Nonlinear Phenomena in Science and Engineering. Ellis Horwood.Google Scholar
Szeri, A. Z. & Al-Sharif, A. 1993 Flow between finite, steadily rotating eccentric cylinders. Preprint.Google Scholar
Versteegen, P. L. & Jankowski, D. F. 1969 Experiments on the stability of viscous flow between eccentric rotating cylinders. Phys. Fluids 12, 1138–1143.Google Scholar
Vohr, J. A. 1968 An experimental study of Taylor vortices and turbulence in flow between eccentric rotating cylinders. Trans. ASME: J. Lub. Tech. F 90, 285–296.Google Scholar
Weinstein, M. 1977a Wavy vortices in the flow between two long eccentric rotating cylinders, I. Nonlinear theory. Proc. R. Soc. Lond. A 354, 441–457.Google Scholar
Weinstein, M. 1977b Wavy vortices in the flow between two long eccentric rotating cylinders, II. Linear theory. Proc. R. Soc. Lond. A, 354, 459–489.Google Scholar
Wood, W. W. 1957 The asymptotic expansions at large Reynolds numbers for steady motion between non-co-axial rotating cylinders. J. Fluid Mech. 3, 159–175.Google Scholar
Zarti, A. S. & Mobbs, F. R. 1980 Wavy Taylor vortex flow between eccentric rotating cylinders. In Energy Conservation Through Fluid Film Lubrication Technology: Frontiers in Research and Design (ed. S. M. Rohde, D. F. Wilcock & H. S. Cheng). The American Society of Engineers, New York.Google Scholar